Logic in Philosophy vs. Mathematical Logic At my university, students majoring in philosophy take a course called "Logic in Philosophy" and there is also a course offered in the Math Department called "Mathematical Logic". There are also different books published in each field (Logic for Philosophy by Sider vs. Mathematical Logic by Enderton and Ebbinghaus).  

Are these two distinct fields? If so, do they share common elements, concepts, and terminology/definitions?
Students majoring in philosophy take a course called "Logic in
  Philosophy" and there is also a course offered in the Math Department
  called "Mathematical Logic". Are these two distinct fields? If so, do
  they share common elements, concepts, and terminology/definitions?


Three points.

(A) Most philosophy students (at least those "majoring" in philosophy) are expected to do at least a Baby Formal Logic course, often in their very first year. Coverage will vary, but they should pick up an understanding of what makes a formally valid argument, of the truth-functional connectives, of the logic of quantifiers and identity, maybe one or two other things too like  familiarity with set notation. This will usually be done very slowly, remembering that most philosophy students have no maths background and many will be symbol-phobic. There may be more formal logic taught later in later years in optional courses, and eventually (though less and less these days) they might be offered a mathematical logic course.

Beginning university mathematics students will at various points, in e.g. courses on discrete mathematics, pick up an understanding of the connectives, of quantifiers, and of course of set notation, as they go along. But they won't usually have a stand-alone course on Baby Formal Logic (which hasn't much mathematical excitement!). So usually the first stand-alone logic course they might be offered is at senior undergraduate/beginning postgraduate course, in which Baby Logic gets covered (again) very fast, but this time with added completeness proofs etc. etc., before going on to other stuff.

So one main difference between an intro course in formal logic for philosophers and the early lectures in a course in mathematical logic for mathematicians is simply speed and level of mathematical sophistication. As noted, by mathematical standards, the combinatorial ideas involved in the philosophers' Baby Formal Logic course are extremely easy and elementary (even if it doesn't always look that way to the students having to take the course!). The same ideas can be covered at a cracking pace at the beginning of a  mathematical logic course.

(B) As I said, there may then be more formal logic offered to philosophers in later years in optional courses -- some of these courses covering various things of particular interest to philosophers, but not so much to (most) mathematicians, like modal logic. The second half of Sider's book indicates the sort of coverage typical of such courses. So here the focus of the coverage of courses in philosophy and mathematics department can indeed diverge.

(C) To take up some remarks in other answers, you need to sharply distinguish "Logic for Philosophers", meaning a treatment of some topics in formal logic that are of interest/use to philosophers, from "Philosophical Logic". The latter is a standard label for an area which is really a department of the broader philosophy of language, interested in questions about e.g. the nature of reference, the nature of logical necessity, the nature of a valid argument, the meaning of ordinary language conditionals, and so on. One technique for addressing such questions is to compare conditionals (to run with that example) in ordinary language with various constructed connectives in artificial languages of logic. So languages of formal logic get into the story as useful objects of comparison. And perhaps that's what unites the ragbag of topics in philosophical logic. But still, the real focus of courses on so-called philosophical logic is in issues that arise informally or pre-formally, not centrally in issues of formal logic.
As a mathematician (not a philosopher), I think that the biggest difference between philosophical logic and mathematical logic is that in mathematics we assume the existence of the most fundamental objects that we're trying to study.

The most common mathematical assumption is the existence of the integers.  They have bizarre properties.  For example, you can always add one to an integer to get a larger integer, i.e, there's an infinite number of them.  Estimates of the number of elementary particles in the visible universe are in the $10^{80}$ to $10^{97}$ range [1], so there might not be $10^{100}$ particles in the universe, yet $10^{100}$ is a well defined integer. $10^{10^{100}}$ is an integer with more digits than there are particles in the universe, yet it "exists", and what's more, I can  construct an integer that's larger than it: $10^{10^{100}} + 1$.  I can even prove that there are infinitely more integers that are larger than $10^{10^{100}}$.

I went to college to learn these proofs.  Aristophanes would approve.  [2]

Of course, integers do not really exist as physical objects.  If we accept the mathematician's claim that they do "exist", then we could do Anselm [3] one better:


Assume the existence of God.
Therefore, God exists.


I call it the "Mathematician's Proof of the Existence of God".

I know of no field of philosophy that so cavalierly assumes things into existence, although all philosophical arguments seem to be based on some claims assumed to be true.  Once these assumptions are clearly stated, then the same process of logical deduction (or induction) can be applied.

The great irony here is that by assuming the existence of physically implausible objects like integers and complex numbers, we can construct theories that explain the behavior of the actual physical universe to previously unimaginable precision.