Contrary to popular belief, mathematicians are not typically any better at adding or subtracting numbers, or figuring the tip on a restaurant bill, than members of any other profession - in fact some of the best mathematicians are notoriously bad at these tasks!
Mathematicians are typically interested in finding and describing patterns which may have originally arisen from problems of calculation, but have now been abstracted to become problems of their own. From much published research work of mathematicians, it may look as if the primary approach of a mathematician is to start with some given assumptions, often called axioms; and then proceed to prove other facts which follow from the assumptions according to exact rules of logic. That, however, is the finished product that gets published; it is not work in progress.
Mathematicians differ from philosophers in that the primary questions of mathematics are assumed (for the most part) to transcend the context of the human mind; the idea that "2+2=4 is a true statement" is assumed to exist without requiring a human mind to state the problem. Not all mathematicians would strictly agree with the above; the philosophy of mathematics contains several viewpoints on this question.
Mathematicians differ from physical scientists such as physicists or engineers in that they do not typically perform experiments to confirm or deny their conclusions; and whereas every scientific theory is always assumed to be an approximation of truth, mathematical statements are an attempt at capturing truth. If a certain statement is believed to be true by Mathematicians (typically as special cases are confirmed to some degree) but has neither been proven nor disproven to logically follow from some set of assumptions, it is called a conjecture, as opposed to the ultimate goal - a theorem which is ultimately true. Unlike physical theories, which may be expected to change whenever new information about our physical world is discovered, mathematcial theories are "static" - once a statement achieves the lauded position of a theorem, it remains true forever.
Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows: