Why was an SS commander dismissed and charged for the mistreatment of Jewish prisoners?

So why did they particularly care that they were being mistreated in a concentration camp?

They didn't. Mistreating and killing Jews was institutionalized and so systemic, that even somebody like Amon Göth, who was openly shooting prisoners from his balcony for fun was not charged. The standing order required SS guards to use their firearm without prior warning in all situations.

Postenpflicht, or subduing an inmate with anything but lethal use of a firearm, was a reason to be transferred out (and not to guard duty in Paris).

He was charged with corruption. A crime against the Nazi state. And while many in his position probably did what he did, he was accused by his own troops, because while he was a brutal butcher to the inmates, he did not turn all friendly and nice to his own people. He was an alcoholic and sadist and he mistreated his own guards and brought his own guards to an SS court for the smallest offences. So basically, they turned him in for the crimes they could find: Embezzlement.

There is a certain amount of logic based on incomplete data at work here. The Wikipedia article has this phrased rather unlucky. It seems that two trials against Göth are mixed up here.

This in part caused by the main source of the WP article, Mr Crowe, who writes:

Five weeks later, the NTN tried Amon Göth, the infamous, sadistic commandant of the Płaszów concentration camp made famous in Schindler’s List. He was accused of various crimes, including murder and the theft of his victims’ property, in Kraków, Płaszów, and Szebnie. The principal witness against Göth was Mietek Pemper, a Jewish inmate who was forced to work for Göth as his personal stenographer and interpreter in Płaszów. Pemper discussed Göth’s arrest by the SS in the fall of 1944 on various charges including theft, murder, and the mistreatment of prisoners.
_{David M. Crowe: "War Crimes, Genocide, and Justice. A Global History", Palgrave Macmillan: New York, 2014, p261.}

But that is already confusing the charges brought up against Göth by his fellow SS-comrades, "mistreatment of prisoners" and the charges brought up against Göth later in his war crimes trial in Poland. "Mistreatment of Prisoners" in SS-language is not the same as mistreament of prisoners in war-crimes trial!

The primary concern the SS had against one of their own was:

Am 6. September geht bei SS-Standartenführer Rudolf Brandt im persönlichen Stab Himmlers ein Telegramm von HSSPF Ost Wilhelm Koppe ein. Koppe reagiert auf eine Anfrage Brandts in Sachen Göth:

„Gegen Hauptsturmführer Göth läuft im Augenblick ein gerichtliches Ermittlungsverfahren wegen Amtsmissbrauchs. Im Laufe des Verfahrens aufgetauchte andere Verdachtsmomente werden gleichzeitig untersucht.“

Translation: On September 6, SS-Standartenführer Rudolf Brandt receives a telegram from HSSPF Ost Wilhelm Koppe in Himmler's personal staff. Koppe reacts to a request from Brandt regarding Göth:

"Hauptsturmführer Göth is currently under investigation for abuse of office. Any other suspicions arising in the course of the proceedings shall be investigated simultaneously."
_{Johannes Sachslehner: "Der Tod ist ein Meister aus Wien," Graz: Styria Premium, 2008; p351. The "other" suspicions were doubtless related to Göth's embezzlements, especially to the wagon in Opawa, full of smuggled goods.}

Mostly concerned about embezzlements, the SS had a long standing order on how to react to that. That video is the edited recording of Himmler's Posener Rede (Poznan speech) from 1943, where he condemns these activities and reminds his listeners that 'killing Jews is fine, robbing them is fine, but that all that is done for the state. And if one SS member would go so far as to steal one cigarette from a Jew while exterminating him that would be punishable by death.'
If the SS did care somehow about treating prisoners, Jews especially, in 'improper ways', what might that have been?

During the clearance of the Tarnow Ghetto Göth misappropriated Jewish property, furniture, furs, clothing, jewellery, tobacco and alcohol, which were later found by the Gestapo in storage, at Brunnlitz. Also at Brunnlitz was Göth’s mistress Ruth Irene Kalder, also known as ‘Majola’.

Göth also supervised the progressive liquidation of the forced labour camp in Szebnie near Jaslo. The liquidation began on 21 September 1943 with the killing of 700 Jewish prisoners who were driven in lorries to a forest in Tarnowiec, 3km from the camp, where they were shot.
This ‘aktion’ was carried out by SS-Hauptscharfuhrer Grzymek and supervised by the commandant Kellermann, acting on orders from Göth.

Göth also prepared under the leadership of Willi Haase, plans for the liquidation of ghettos in Bochnia, Rzeszow and Przemysl.

On 13 September 1944 Göth was arrested by the SS und Polizeigericht Vl (Police Court) in Krakow for large-scale fraud.

Göth was also interrogated by the Sicherheitspolizei (Security Police) for giving information to the engineer Grunberg about the liquidation of the Krakow Ghetto. Grunberg a German was sympathetic to the Jews and closely associated with Stern, Pemper and Schindler. He passed the information on to Schindler, who in turn warned the ghetto leaders.
_{Holocaust Education & Archive Research Team: Amon Göth}

Beating or shooting prisoners is in the view of the SS no ground for grudges. Although the theoretical argument might be made that "not providing adequate food" might be a sign of mismanagement of a camp where the prisoners had to do important work for the war effort, this is obviously not followed closely in any of the camps.

Aber es ging leider nicht nur um Göths Schwarzmarktgeschäfte. Der SS-Richter, der Pemper vernahm, wußte von Oberscharführer Lorenz Landsdorfer, daß der Kommandant von seinem jüdischen Schreiber die Direktiven und Aktionspläne hatte tippen lassen, die im Falle eines Partisanenangriffs auf das Lager in Kraft treten sollten. Um ihm zu zeigen, nach welchem Schema diese Pläne zu tippen seien, habe Göth ihm sogar Kopien von Plänen gezeigt, die für andere Konzentrationslager ausgearbeitet worden waren. Der Untersuchungsrichter ließ Pemper verhaften, denn er war äußerst beunruhigt darüber, daß diese geheimen Pläne einem jüdischen Häftling zur Kenntnis gekommen waren.
Translation: But it wasn't just Göth's black market business. The SS judge, who heard Pemper, knew from Oberscharführer Lorenz Landsdorf that the commander had his Jewish scribe type in the directives and action plans that were to come into force in the event of a partisan attack on the camp. In order to show him according to which scheme these plans are to be typed, Göth had even shown him copies of plans that had been drawn up for other concentration camps. The investigating judge had Pemper arrested because he was extremely concerned that these secret plans had come to the attention of a Jewish prisoner.
_{Thomas Keneally: " Schindlers Liste", Goldmann: München, 1983.}

One of the mistreatments was letting the prisoners come too close, also in terms of knowing too much about the internal workings of and plans for Göth's camp as well as other camps.

The German Wikipedia is a bit clearer on the subject matter:

Er behandelte auch SS-Untergebene hart und brachte sie wegen kleinster Vergehen vor ein SS- und Polizeigericht. Zudem betrieb er Schwarzmarktgeschäfte. Dies führte dazu, dass er von SS-Untergebenen wegen Unterschlagung von Reichseigentum (nach NS-Recht fiel das konfiszierte Eigentum der jüdischen KZ-Gefangenen dem Deutschen Reich zu) angezeigt wurde. Hingegen wurde die Ermordung von KZ-Häftlingen durch die NS-Justiz nicht geahndet, sondern forciert (vgl. sogenannte „Postenpflicht“).
Translation: He also treated SS minions hard and brought them before an SS and police court for minor offences. He was also in the black market. This led to him being reported by SS subordinates for embezzlement of Reich property (under Nazi law, the confiscated property of the Jewish concentration camp prisoners fell to the German Reich). On the other hand, the murder of concentration camp prisoners was not punished by Nazi justice, but enforced. (Compare Postenpflicht)
_{Wikipedia: Amon Göth}

Adding to those misdeeds within the sphere of labour- and concentration camps are the behaviours on display, Göth was also seen very unfavourable by his fellow officers. Not only was he overly harsh in his treatment of those ss-men he was in charge of, he was also conspiring and intriguing against those who were on his level of command.

It was his subordinates who turned him in….He was well known for his brutality and for taking advantage of both prisoners and subordinates.”87 The egomaniacal commandant had treated even his own SS men very harshly. In return, they envied and feared him. Göth’s openly conducted black-market deals were especially maddening to his subordinates, since he would have them hauled before an SS and police court for the slightest offense. On Göth’s orders, I wrote up several such indictments. Those convicted would have had to serve their sentences after the war. I remember one of them almost whining, “You’ll all be celebrating the end of the war and I’ll have to go to jail.” One time Göth sent the Jewish mechanic Warenhaupt to town to get some parts for his private BMW, and SS-Rottenführer Krupatz—an older, easy-going fellow—was detailed to accompany him. Warenhaupt, an athletic skier from Zakopane, was able to escape. At a building with a front and back entrance, he told Krupatz, “The Pole with the replacement parts lives in here. But he’ll only give them to me if I go in alone. Wait here for me.” Krupatz was fooled and had to return to camp without Warenhaupt. Predictably, Göth turned in Krupatz for “negligent freeing of a prisoner” and “abetting an escape.” Later on, I learned that regulations actually required there always to be two SS men accompanying a prisoner on an errand to town. This turned out not to have been an isolated incident, and at last the fury of the SS men against Göth reached its boiling point. They got together and composed a complaint that essentially said, “Göth is living like a pasha while our soldiers are dying on the eastern front.” This complaint eventually reached the desk of the SS judge Dr. Morgen, the transcripts of whose hearings were later used even by the military tribunal in Nuremberg. Abruptly, Göth’s career in the SS was over.[…]
Was it possible, I thought to myself, that by the end of 1944, there was an SS judge who still didn’t know what the camp policies of his country were? Who didn’t know that no legal sentence was necessary to send Jews to their death? Then he asked if there was any possibility I would ever be assigned to a work detail outside the camp. Apparently, he was afraid I would be able to carry secrets to the outside. I assured him truthfully that I had never been employed outside the camp, which was confirmed by SS-Hauptscharführer Schupke.[…]
It was a relief that all the magistrate’s remaining questions revolved around Göth’s personal machinations and private schemes. Göth had actually toyed with the idea of purchasing a country estate and a banking house. He had once asked me to prepare a list of questions he would need to ask when negotiating such purchases, and had also discussed the matter with his father. After I showed the magistrate the binder with the documents proving Göth’s purchase plans, I was finally allowed to return to the camp. It was September 27, 1944, two weeks to the day after Göth’s arrest and the eve of Yom Kippur, the end of the Jewish High Holy Days.
_{Translated from Mieczysław "Mietek" Pemper: "Der rettende Weg. Schindlers Liste – Die wahre Geschichte. Aufgezeichnet von Viktoria Hertling und Marie Elisabeth Müller. Hoffmann und Campe: Hamburg, 2010."}

This is an almost universal fall from grace that is evidenced in that an official SS magazine accused him, in his function of head of the publishing house his father founded, of using "gangster methods" in selling books:

_{From Kapitel 27 - Prozess und Hinrichtung}

One important aspect in this case is that the pluri-centric nazi-system has some surprises behind the simple stories. Here it is that the prosecutor of Göth is Konrad Morgen, an SS-judge that rose and fell in popularity himself repeatedly during his career in the NS system:

After requesting a transfer, Morgen was instead dismissed by Himmler, ostensibly for acquitting an SS officer of the racial crime of sexual relations with an alien race, but also perhaps for meddling in Himmler's affairs. He was punished by being sent to the Wiking Division on the Eastern Front. However, in mid-1943, Himmler recalled Morgen to investigate and prosecute corruption in the concentration camp system, which had become rampant, as reflected in Himmler's notorious Posen speeches.

Despite Morgen being quite a 'complicated character' and his actions being labeled as "difficult" to compare with others, even in hindsight, his actions were decisive in persecuting Göth, as most other SS officers who had trouble of a similar kind while still being active in the SS before the conclusion of the war.

Morgen’s sentiments – […] — were invested in his professional role as an SS judge, a role in which he immersed himself with a passion. But that role was itself conflicted. On the one hand, an SS judge applied the same body of law as civil and military judges: there was no distinct SS legal code. On the other hand, the SS judge was supposed to apply the civil and military codes in a distinctive manner appropriate to members of the SS. Morgen therefore had a second allegiance not incumbent on other judges applying the same body of law. His second allegiance was to the value system of the SS.[…]
After viewing the gas chambers of Auschwitz, he could still be shocked by SS men fraternizing with Jewish girls. He condoned hard treatment of subject peoples in the East. He referred to mass execution by shooting as “the old, tried method”—whatever that may have meant. When he returned to Cracow at the end of 1944, he was gratified to hear that his reputation as a hanging judge (Blutrichter) had preceded him, striking fear into the hearts of prisoners.
_{Herlinde Pauer-Studer & J. David Velleman: "Konrad Morgen. The Conscience of a Nazi Judge", Palgrave Macmillan: New York, 2015, p121.}

Conclusion: "Why was an SS commander dismissed and charged for the mistreatment of Jewish prisoners?"

Behaviour such as that of Göth was widespread, including black market activities and his treatment of prisoners. He got into the wheels because he treated both prisoners and his own underlings badly. He was accused, investigated for embezzlement and abuse of office, but then only charged, found guilty and dismissed for embezzlement and economic crimes alone.
He was first charged with a laundry list of misdeeds, as was almost customary in cases of denunciation. These accusations included irregularities in running "his" camp but were mostly centered around his embezzlements and illegal activities concerning valuable goods. In NS-terminology, prisoners do not fall into this category. That he was reported for this kind of activity does not mean that his superiors cared for all of the points brought forward.
Göth was not punished until after the war for his inhumane treatment of prisoners.

He was stealing the Jews' property, which belonged to the state, which in practice meant the SS. That meant he was showing obvious corruption, in an organisation that tried very hard to present an image of impeccable honour and loyalty, but was actually corrupt to the core.

There may well have been written regulations about the humane treatment of prisoners, but if so, they were intended purely for showing to the Red Cross and the neutral powers, so that the SS could claim they were doing things properly. I doubt they expected anyone to believe any of this, but it made it easier for inspectors to pretend that things were in order, rather than objecting and insisting on on-site inspections. The SS were good at making it easier for people to be corrupt.

The hypocrisy was entirely institutionalised, so charging Göth with violating regulations that he was normally required to ignore is just a way of having more charges. The charge of allowing prisoners access to camp personnel records was probably genuine: it presented a risk of prisoners finding a way to establish personal contact with guards and make the guards see them as people. The SS had learned that they had to dehumanise their prisoners, to avoid the guards suffering psychologically when required to kill them.