How are Prasangika and Svātantrika different?

First, the Gelug and the Nyingma presentations of the Prasangika system vary (in particular with respect to other-emptiness / self-emptiness). My answer is from the Gelug standpoint.

Second, I am confident that the following references would answer your questions:

1. Editor's introduction to The Adornment of the Middle Way: Shantarakshita's Madhyamakalankara with Commentary by Jamgon Mipham,
2. Editor's introduction to Introduction to the Middle Way: Chandrakirti's Madhyamakavatara with Commentary by Ju Mipham, and
3. The Svatantrika-Prasangika Distinction: What Difference Does a Difference Make?

Prasangika give a short definition of conventional truth:

An object found by a mind distinguishing conventionalities.

While the short definition of ultimate truth is:

An object found by a mind distinguishing the ultimate.

Although the Svatantrika definitions differ, these two help understand the issue at hand. The definition mentions what consciousness has one or the other truth as its object: in one case, "a mind distinguishing conventionalities", in the other "a mind distinguishing the ultimate." It is important to understand what a conventional valid cognizer and an ultimate valid cognizer are, because they apprehend different objects. A conventional valid cognizer apprehends conventional truths while the mind analyzing the ultimate apprehends ultimate truth, the lack of inherent existence. Nihilist think that, since a mind analyzing the ultimate does not find conventionalities (i.e. all it finds is the absence of inherent existence, emptiness) then conventionalities are utterly non-existent. This is the reason why Khedrup Je reminds us: not seeing sounds does not refute the existence of sounds... in the same way, not finding conventional truths with a mind of ultimate analysis does not refute the existence conventional truths. It is not because an Arya in meditative equipoise on emptiness does so in a non-dualistic manner, not apprehending any conventionality, that conventionalities are refuted. Conventional truths and ultimate truths are just the objects of two distinct cognizers, just like smells and sounds are the objects of two distinct consciousnesses.

Both Svatantrika and Prasangika assert that ultimate truth - the absence of inherent existence - is not found by any conventional valid cognizer: it is found only by the pristine wisdom realizing emptiness. This wisdom is a mind that we also call "mind distinguishing the ultimate", "mind of ultimate analysis", "mind analyzing the ultimate." Svatantrika posit that phenomena are empty of inherent existence ultimately because the mind of ultimate analysis apprehends the lack of inherent existence, but they are not empty of inherent existence conventionally because such emptiness is not found by any conventional valid cognizer. Khedrup Je says:

Therefore, when [Svatantrika] explain that entities are natureless, it should be taken in the context of referring to an ultimate [nature, and should not be taken as denying that things exist from their own side]

Prasangika disagree with this assertion. They posit that inherent existence is utterly non-existent. Therefore, phenomena do not exist inherently, be it conventionally or ultimately. Emptiness of inherent existence is not found by any conventional valid cognizer but this does not entail that phenomena are empty only ultimately and not conventionally. According to Prasangika, phenomena are empty of inherent existence even conventionally, since inherent existence is a mode of existence that is refuted by ultimate analysis. Since emptiness of inherent existence is apprehended by an ultimate valid cognizer, its opposite – inherent existence – must be utterly non-existent. Another way of saying is that inherent existence is not an object of knowledge of even conventional valid cognition. Therefore, no phenomena exist inherently in any way.

Both Jetsün Mipham and Geshe Dreyfus suggest that this distinction was overly emphasized by Prasangika while needed not be. It is the point of Geshe Dreyfus's book: What Difference Does a Difference Make? In the introduction to Mipham's commentary to Shantarakshita, it also says:

Those who, through practice associated with the view of the concordant ultimate, thus attain the experience of the ultimate truth in itself may be called either Prasangika or Svatantrikas depending on the way they make or do not make assertions in subsequent attainment. But one should know that in terms of their realization there is no difference between them. They are both in possession of the wisdom the Aryas.

Prasangika claim, however, that the object of negation asserted by Prasangika is not the same as that of the Svatantrika. Asserting that anything exist inherently in any way (as the Svatantrika do) is untenable. Therefore, according to Prasangika, only a proponent of Prasangika tenets is “in possession of the wisdom of the Aryas”. Indeed, we often believe that all Superiors (Aryas) must be Prasangika by tenets.

The thing is that Svatantrika posit a conventional analysis while Prasangika do not. Since they posit the entity of conventional valid cognizer differently, what these cognizers apprehend must differ as well. Prasangika argue that no conventionality is an object of analysis, but an object apprehended by a conventional valid cognizer that does not investigate the mode of existence of anything. The introduction to Mipham's commentary to Shantarakshita reads:

[Some] think that to affirm a tenet that investigates conventional phenomena is incompatible with the Prasangika stance, which is to accept phenomena as they are, without analysis, according to the general consensus.

This is because conventionalities – conventional truths – are just that: conventions. They are not to be refuted or investigated. Once, in a class I was attending, a student asked our teacher “Since I only see John's body, and the person is not intrinsically one with her body, then it means I have never seen John, doesn't it?” It is due to this type of misunderstanding that Prasangika refute any conventional analysis. It shows just how important it is to understand how conventional truths are objects of conventional cognizer, not of minds analyzing the ultimate. As soon as one engages in analysis, he leans towards the direction of realizing emptiness, and the mind realizing emptiness does so in a non-dualistic manner, it does not apprehend conventional truths.

On the other hand, Svatantrika posit that one can analyze the mode of existence of phenomena by way of a mind of conventional analysis, and thus refute that phenomena exist in certain ways conventionally. For instance, Svatantrika-Yogacara posit that external establishment can be refuted with a conventional analysis. Thus, they claim that phenomena lack external establishment conventionally, do not lack inherent existence conventionally, lack inherent existence inherently. They lack external establishment conventionally because a conventional valid cognizer can refute external establishment; they do not lack inherent existence conventionally because the emptiness of inherent existence can not be found by way of conventional analysis; they do lack inherent existence ultimately because the emptiness of inherent existence is found / realized by way of a mind analyzing the ultimate.

The introduction to Mipham's commentary to Shantarakshita reads:

It must be said, however, that in the context of pramana, or valid cognition, applied on the conventional level, [according to Svatantrika] it is quiet acceptable to say that phenomena exist according to their characteristics or that they are established by valid cognition and so forth.

Prasangika say, on the other hand: (1) there is no such thing as conventional analysis (2) inherent existence is utterly non-existent even if its absence is found only by a mind of ultimate analysis, because inherent existence and its emptiness are opposites. It is important to understand what conventional valid cognition and ultimate valid cognitions are to understand what their respective objects are. In the introduction to Mipham's commentary to Shantarakshita, it says:

The Prasangika texts refute 'true existence' (bden grub) and 'existence according to characteristics' (rang mtshan nyid kyis grub pa) indiscriminately. But when one makes a distinction between these expressions – as the Svatantrikas do (refuting true existence on the ultimate level and asserting existence according to characteristics on the conventional level) – one must distinguish, if one is not to confuse the issue, the two kinds of reasoning and their respective spheres.

Additionally, according to Prasangika, if a phenomena was inherently existent, then it would have to appear the way it exists (i.e. it would have a truly existent phenomena).

According to Svatantrikas, however, conventional truths are empty of true existence (i.e. they do not exist the way they appear) but are not empty of inherent existence conventionally. When Prasangika say that a phenomena does not exist inherently, they mean that if you look for it with a mind analyzing the ultimate, you do not find it. Svatantrika mean something else, since they posit both that (1) they exist inherently and (2) they do not withstand ultimate analysis. In the introduction to Mipham's commentary to Shantarakshita, it says:

[According to Madhyamika-Svatantrika-Yogacara] if conventional phenomena were assessed from the standpoint of ultimate valid cognition, they would not be even slightly established thereby.

This helps understand the Svatantrika position that conventional truths do not exist inherently ultimately but do exist inherently conventionally.

How do conventional phenomena appear, according to Svatantrikas? They appear as:

1. existing by way of their own characteristics
2. not being imputed by mind

While they are, both:

1. existing by way of their own characteristics, and
2. imputed by mind.