@Mauro ALLEGRANZA beat me to it with an excellent answer but since i had already drafted this here's another take:
The answer is "neither".
It is definitely not the claim that mathematics and logic are the same thing. The central concern of logic, on most accounts, is the notion of logical consequence and/or valid inference. You commonly find words to this effect at the beginning of introductory texts. It's a lot harder to say what the central concern of math is. Historically notions like quantity, magnitude, number, etc have been offered but math covers so much conceptual ground it's hard to pick out one thing common to all but not trivial. Logical consequence by contrast is not only non-trivial, it remains somewhat mysterious. Compare the notion of computability, which remained mysterious until Turing came along and provided a model that is universally considered to be satisfactory both formally and intuitively. Nobody has yet managed to do that for the informal, intuitive concept of logical consequence.
Note that there is no prima facie reason to think that our intuitive notion of logical consequence has anything to do with mathematics.
What about "the logic of mathematical reasoning"? Mathematical reasoning, by definition, is what mathematicians do when they "do" mathematics, and it is informal. It is now possible to publish formal proofs of mathematical results, but almost nobody does this and some think it is a bad idea. Mathematical argumentation may be highly disciplined, but it is not formal. However mathematical logic is formal. You could argue that ML is an attempt to formally express the informal reasoning of working mathematicians, but that's something very different than offering a "logic of mathematics". The latter suggests ( to me at least) an attempt to give an explanation (in some formal system, which is itself a mathematical object) of how informal mathematical reasoning works, which is really a philosophical project. ML does not explain anything, it's just a technology.
It also depends on what you mean by "mathematics". Classical or intuitionistic? Most texts on ML that I've come across implicitly assume that Classical Mathematics (which relies on truth-conditional semantics) is the only game in town, which is not the case.
I personally think of mathematical logic as mathematicized logic: treatment of logical forms as mathematical objects, without claiming that logic is reducible to mathematics. So it's more of a methodological concept.