Savage used as a noun is an offensive word for sb who belongs to a people that is simple and not developed, like primitive people, more like animals, less human. Wild has a wider scope of meaning. When opposite to domestic, it means living or growing in natural conditions. Or if you describe a person wild, you mean this person is lack of discipline or control, for example, a wild child. Or it can be used to describe a state of ecstasy. The two words are confusing because they both have a meaning of rude or violent. But the former is because they are not civilized，thus beastly. And the latter is just because they are too excited or lack of discipline. Of course a dictionary can offer you the most detailed information. Pls take my answer with a grain of salt. And good question.

'savage' used to be used to describe groups of people that are not very 'developed'. However that use is quite dated. It is the sort of description you would have heard 50-100 years ago. Now, it means violent and dangerous. That could be either an innate characteristic, or because of the effect of drugs, or anger. 沈梦丹 gave a reasonable description of wild. The words are similar, and someone that is wild in the sense of undisciplined could be savage, but not necessarily. Most people that are wild still have enough self control to not hurt other people.