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?
 A: 
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. 
A: The main difference between "Logic in Philosophy" and "Mathematical Logic" is that in the former case logic is used as a tool, while in the latter it is studied for its own sake.
A Logic class in a Philosophy degree will usually cover sentential, predicate and finally first-order logic (by order of increasing complexity and natural way of learning). At this point, you're able to give a formal expression of philosophical arguments, reason in terms of truth-value, validity and soundness, and you can detect and explain fallacies.
This is why (elementary, first-order) Logic is such an important tool in Philosophy: you need to be able to understand why the text you're reading is convincing, what the author's thoughts are, how he connects them together, how he draws conclusions from premises and so on. A frequent extension is modal logic, where you introduce the notions of necessity and possibility, and which is the basis for deontic logic - the language of Ethics.
On the other hand, "Mathematical Logic" uses the power of logic to formulate concepts in an unambiguous language where you can perform operations in order to lay the foundations of mathematics. If you study mathematical logic, you'll be introduced to much more complex types of logic, with which you'll then be able to define rigorously concepts such as theorems, deductions, axioms, sets, functions, graphs, representations, etc. - all of which form the basis for a meta-language that you know as "mathematics".
A: i will attempt an answer in point-by-point kind of way, hope it is useful


*

*Mathematical logic is symbolic and formal, Philosophy logic is more informal, more natural language oriented

*As a result not all the forms of logic in Philosophy can be formaliserd mathematicaly, and vice-versa mathematics can formalise other notions of logic not used in philosophy (e.g toy models etc.)

*There is an interesting topic in the foundations of mathematics (or mathematical philosophy) which exactly tries to study the connection (or difference) between the philosophical logic and mathematical logic and whether logic is the foundation of mathematics or vice-versa (many schools here with sometimes major differences, i.e formalism, intuitionism, mathematical platonism, mathematical realism, philosophy of probability and inference, etc)


update
To address the comments by @Michaelangelo here are some examples, with references, of logic(s) in philosophy different from analytical logic(s):
Dialectical Logic(s) vs Analytical Logic(s)


*

*Pre-Logic, Formal Logic, Dialectical Logic, marxists.org

[..]Contrary to formal logic, the law of dialectical logic is that
  everything is mediated therefore everything is itself and at the same
  time not itself. "A is non-A." A is negated.



*Formal Logic and Dialectics, marxists.org

[..]Ilyenkov explains in his essay on Hegel, that Hegel's revolution in
  logic was effected by widening the scope of logic and the field of
  observation upon which the validity of logic could be tested, from
  logic manifested in the articulation of propositions to the
  manifestation of logic in all aspects of human practice.



*Dialectics and Logic

[..]Dialectics and formal logic are sometimes posed as two contrasting forms of reasoning. In this contrast, formal logic is appropriate for
  reasoning about static properties of separate objects involving no
  interaction. To deal with change and interaction it is necessary to
  use the dialectic approach.



*Dialectical logic, wikipedia

[..]Rather than the abstract formalism of traditional logic, dialectical
  logic was meant to be a materialistic examination of concrete forms:
  The logic of motion and change. A logic that is a statement about the
  objective material world.

Intuitionism vs Formalism
While being a school of mathematical philosophy, Intuitionism is not analytic, nor its logic.


*

*LEJ Brouwer, Lectures on Intuitionism, marxists.org

FIRST ACT OF INTUITIONISM
Completely separating mathematics from mathematical language and hence
  from the phenomena of language described by theoretical logic,
  recognising that intuitionistic mathematics is an essentially
  languageless activity of the mind having its origin in the perception
  of a move of time. This perception of a move of time may be described
  as the falling apart of a life moment into two distinct things, one of
  which gives way to the other, but is retained by memory. If the twoity
  thus born is divested of all quality, it passes into the empty form of
  the common substratum of all twoities. And it is this common
  substratum, this empty form, which is the basic intuition of
  mathematics.

Informal Logic(s) vs Formal Logic(s)
Formal Logic(s) do not make full use of natural language constructs like subtleties, ambiguities and shades of meaning. 


*

*Informal Logic and the Dialectical Approach to Argument 

INFORMAL LOGIC: ORIGINAL CONCEPTIONS
In reflecting on the origins of informal logic, Johnson and Blair
  (2002, pp. 340-352; cf. 1980, p. 5) describe it as arising in the
  context of three stream s of criticism to the existing academic logic
  program.  First the pedagogical cr itique challenged that the tools of
  logic should be applicable to ever yday reasoning and argument of the
  sorts used in political, social and practical issues. Second the
  internal critique challenged the adequacy of  existing tools of logic
  in evaluating everyday argument.  Specifically rejected was the
  logical idea of soundness as either a necessary or a sufficient
  criterion for the goodness of arguments, as well as a formalistic
  understandi ng of validity. Finally, the empirical critique challenged
  the ideas that formal  deductive logic can provide a theory of good
  reasoning, and that the abil ity to reason well is improved by a
  knowledge of formal deduction. As well, Johnson and Blair (2002, p.
  355) associate the genesis of informal  logic with a renewed interest
  in the informal fallacies which were also inadequately treated in
  traditional logic programs of the time.

While there are philosophical currents that make full use of such features and usualy also have a "poetic" flavor (poetry in philosophy or philosophy in poetry). As such make extended use of poetic tools involving natural language meaning and semantics in complex ways not formalised in formal logic (lets say informal logic).
Works by great philosophers of the past and present who were also poets (or used such an approach to philosophy) serves most appropriate, like Plato, Hegel, Heraclitus, Borges and so on.
A: 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.
