I know that both of these words mean "proof", but I can't figure out what the difference is, or if there's any at all. Can anyone explain to me the difference in nuance between these two words?
Broadly speaking, 証明 is proof and 証拠 is evidence. That is to say, you provide evidence in order to prove something. Accordingly, 証明 is both a noun and a verb, but 証拠 is not a verb, or in other words you can 証明する, but you cannot 証拠する.
This dictionary entry for 証明 makes the relationship pretty clear:
Emphasis added. There are legal uses that follow the general English pattern, for example a judge will use evidence, documented evidence, to make a decision, to judge if that evidence sufficiently proves the claim.
As stated in the other answer, 証明 also refers to a mathematical proof (論証).
証明 is used for conceptual proof, e.g. in the mathematical sense.
証拠 is substantial proof in the sense of evidence, like evidence used in court, e.g. as in the phrase