The gemara has a discussion whether kofer is a "kapara" - atonement, or a k'nas - penalty. If kofer is a penalty, the differences between kofer and damim - compensation, are very clear (i.e. mo'deh b'knas patur). But if kofer is a kapara (the gemara 41b is explicit that admitting would be chayev, and even Raba 43a would only patur because of a technicality that the ox isn't killed), what are the differences between kapara payments and compensation? If kofer is the value of the damager, it is clearly very different than compensation which is the value of the nizuk, but if kofer is the value of the nizuk (machlokes 40a), what is the difference?
1. Rashi (d.h. lo damim) indicates that if one is an o'nes in that he has no money, he is not in violation of anything for not paying a "damim" type payment, but if it is a "kapara" he would remain susceptible to punishment until he pays.
2. Tosafos (d.h. mai) - If the mazik dies, he is not in need of paying for his kapara, because after one dies he is not in need of this kapara, just as korbanos that he failed to bring in his lifetime, need not be brought in his death. This answer seems to be a big chiddush. We find that a person can accrue zechuyos even after he is dead (such as when his children say kaddish, or a d'var torah is said in his name), so why can't he obtain kappara from a payment after his death? Tosafos seems to understand that the kapara is not achieved by the nizuk's inheritors receiving, rather it is achieved by the mazik paying. Once the mazik is dead, his estate no longer belongs to him so he would not be paying and therefore cannot receive atonement.
3. Tosafos (d.h. mai) - The nizuk's inheritors cannot be mochel on receiving kofer, because until the family of the nizuk receive the kofer, the mazik goes without kapara. It would seem to follow that since the mazik is entitled to a kapara through paying kofer, the nizuk's inheritors have no right to refuse accepting the kofer.
4. Gemara 40a - There is a possibility that an ox owned by partners that kills a person, each one would have to pay full kofer, rather than dividing the payment.
5. Gemara 40a - There is also a possibility that for kofer it is assumed that the person will pay to receive atonement so it is not necessary to take a collateral from him, but for a compensation payment we take a collateral.
6. Ra'avad (mentioned in Rashba) suggests that perhaps kofer is paid based on erchin value rather than real value in the market, but the rashba argues.
7. Ra'avad (in Shita) suggests that for a ba'al chov we are mesadrin - we leave behind certain essentials when we collect from him, but for a kapara we would not be mesadrin.
8. Me'iri explains that we are only bound to come up with a distinction between kofer and damim, if we interpret the gemara that for killing a person one can sometimes be patur from kofer but have to pay damim. The meiri rejects this peshat in the gemara and therefore says we do not need to come up with any distinction between kofer and damim, because in a case of kofer, one is never chayev damim.