How does Ajita Kesakambali compare to the Buddha?

Reading the first quote in the OP, it seems to me that Ajita Kesakambalin is teaching that there's no continuation into the hereafter ... no after-life, no heaven (and therefore no reason for generosity, if e.g. the reason for generosity is to earn as reward an after-life in heaven). It's difficult for me to see/understand how a belief in "annihilation" (at death) could cause him to also say there is no "this life" (as well as "there is no next life"). Perhaps it's an (imperfect) attempt to teach the great king about emptiness, and about not attaching to results.

Conversely the Buddhist Dhamma is meant to be akaliko e.g. pertaining to the here-and-now ... intentions and actions are consequential in this life.

Perfect right view of buddha is:

There are fruits, and effects of good or bad actions, causes (paṭiccasamuppada and paṭiccasamuppanna). Elements such as 89 consciousnesses, 52 mind factors, and 28 matters, are causes and effects of each others. And one effect is arised by causes, so if when causes will change to be impossible to make that effect arise, that effect follow vanish or impossible to be, to arise, too (anicca dukkha anatta).

But the other teachers have not perfect right view like the buddha. Such as ajitakesakambala's view that his elements, causes, have not their effects, fruits and results.

I am sorry. I write this answer on mobile phone. So it maybe not too perfect.

The Buddha's teachings are different from what Ajita Kesakambali taught.

The Buddha realized this (from MN 19):

"When the mind was thus concentrated, purified, bright, unblemished, rid of defilement, pliant, malleable, steady, & attained to imperturbability, I directed it to the knowledge of recollecting my past lives. I recollected my manifold past lives, i.e., one birth, two... five, ten... fifty, a hundred, a thousand, a hundred thousand, many eons of cosmic contraction, many eons of cosmic expansion, many eons of cosmic contraction & expansion: 'There I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure & pain, such the end of my life. Passing away from that state, I re-arose there. There too I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure & pain, such the end of my life. Passing away from that state, I re-arose here.' Thus I remembered my manifold past lives in their modes & details. .....

"When the mind was thus concentrated, purified, bright, unblemished, rid of defilement, pliant, malleable, steady, & attained to imperturbability, I directed it to the knowledge of the passing away & reappearance of beings. I saw — by means of the divine eye, purified & surpassing the human — beings passing away & re-appearing, and I discerned how they are inferior & superior, beautiful & ugly, fortunate & unfortunate in accordance with their kamma: 'These beings — who were endowed with bad conduct of body, speech & mind, who reviled the Noble Ones, held wrong views and undertook actions under the influence of wrong views — with the break-up of the body, after death, have re-appeared in the plane of deprivation, the bad destination, the lower realms, in hell. But these beings — who were endowed with good conduct of body, speech, & mind, who did not revile the Noble Ones, who held right views and undertook actions under the influence of right views — with the break-up of the body, after death, have re-appeared in the good destinations, in the heavenly world.' Thus — by means of the divine eye, purified & surpassing the human — I saw beings passing away & re-appearing, and I discerned how they are inferior & superior, beautiful & ugly, fortunate & unfortunate in accordance with their kamma.

The Buddha taught his followers to reflect in this way (from AN 5.57):

“This noble disciple reflects thus: ‘I am not the only one who is the owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s origin, kamma as one’s relative, kamma as one’s resort; who will be the heir of whatever kamma, good or bad, that one does. All beings that come and go, that pass away and undergo rebirth, are owners of their kamma, heirs of their kamma; all have kamma as their origin, kamma as their relative, kamma as their resort; all will be heirs of whatever kamma, good or bad, that they do.’ As he often reflects on this theme, the path is generated. He pursues this path, develops it, and cultivates it. As he does so, the fetters are entirely abandoned and the underlying tendencies are uprooted.

From SN 12.19:

“Bhikkhus, for the fool, hindered by ignorance and fettered by craving, this body has originated. For the fool that ignorance has not been abandoned and that craving has not been utterly destroyed. For what reason? Because the fool has not lived the holy life for the complete destruction of suffering. Therefore, with the breakup of the body, the fool fares on to another body. Faring on to another body, he is not freed from birth, aging, and death; not freed from sorrow, lamentation, pain, displeasure, and despair; not freed from suffering, I say.

From SN 44.9:

“And, Master Gotama, when a being has laid down this body but has not yet been reborn in another body, what does Master Gotama declare to be its fuel on that occasion?”

“When, Vaccha, a being has laid down this body but has not yet been reborn in another body, I declare that it is fuelled by craving. For on that occasion craving is its fuel.”

Also, please note that the term "kāya" has been used in the sense of physical body for example in SN 22.56 (although I know that it can be used to mean group or collection, when combined with other things):

eye, ear, nose, tongue, body, and mind consciousness.
cakkhuviññāṇaṃ, sotaviññāṇaṃ,ghānaviññāṇaṃ, jivhāviññāṇaṃ, kāyaviññāṇaṃ, manoviññāṇaṃ

Pali words for body apart from "kāya" has been used in this context, for example "sarira" in Dhammapada 400:

Akkodhanam vatavantam
silavantam anussadam
dantam antimasariram¹
tamaham brumi brahmanam

Verse 400: Him I call a brahmana, who is free from anger, who practises austerity, who is virtuous and free from craving, who is controlled in his senses and for whom this body (i.e., existence) is the very last.

Footnote 1. antimasariram: lit., one who has the last body. This is his last body because he will not be reborn; he is an arahat.

From SN 15.3:

"This is the greater: the tears you have shed while transmigrating & wandering this long, long time — crying & weeping from being joined with what is displeasing, being separated from what is pleasing — not the water in the four great oceans.

"Long have you (repeatedly) experienced the death of a mother. The tears you have shed over the death of a mother while transmigrating & wandering this long, long time — crying & weeping from being joined with what is displeasing, being separated from what is pleasing — are greater than the water in the four great oceans.

"Long have you (repeatedly) experienced the death of a father... the death of a brother... the death of a sister... the death of a son... the death of a daughter... loss with regard to relatives... loss with regard to wealth... loss with regard to disease. The tears you have shed over loss with regard to disease while transmigrating & wandering this long, long time — crying & weeping from being joined with what is displeasing, being separated from what is pleasing — are greater than the water in the four great oceans.

"Why is that? From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on. Long have you thus experienced stress, experienced pain, experienced loss, swelling the cemeteries — enough to become disenchanted with all fabricated things, enough to become dispassionate, enough to be released."

Also, please note that since some people do not accept the Digha Nikaya as genuine teachings of the Buddha (please see this question), I have quoted from all the other nikayas instead. Dhammapada is part of KN.

That quote from Ajita Kesakambalin is very similar to what Buddha taught, especially if we view his "no this, no that" as pointing that those things are selfless, devoid of substance.

The difference is that Buddha looked in wider context, not limited by the scope of the ordinary observation. For example, if something has ended as material thing, it doesn't mean its influence ended; likewise, personal human life doesn't end in borders of the living body neither spatially nor temporally. Life of the Buddha and its influence until today all over the world is an example of that. Therefore, limits of seeing "one life" are illusory, and so Buddha spoke of other lives etc. - beyond the illusory limits of ordinary consciousness.

his mental fabrications have ceased & subsided

This refers to the mental activity in this body. It doesn't mean that there's no next life after death, unless we are talking about an Arahanth. This is why having a good teacher is important. If someone with materialistic views try to interpret this sutta by himself, he might conclude that it all ends after death.