Arithmetic is simply the calculation/computational discipline. In terms of my particular interest in mathematical cognition, I consider arithmetic one of the building blocks necessary to mathematics.
Mathworld (a really great online source for short answers to math questions and just fun reading) defines arithmetic as "the branch of mathematics dealing with integers or, more generally, numerical computation. Arithmetical operations include addition, congruence calculation, division, factorization, multiplication, power computation, root extraction, and subtraction. " (http://mathworld.wolfram.com/Arithmetic.html)
Balancing your checkbook is arithmetic. Having a degree in mathematics does not mean that I'm any better (probably significantly worse) at that than the average person.
Mathematics is more difficult for me to define because it covers many areas so back to Mathworld for a definition. "Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined." (http://mathworld.wolfram.com/Mathematics.html) Things like logic, which is essential to proofs, figure prominently in the study of mathematics.
Mathworld's main index lists the following topics and serves as a fairly good example of what is included in mathematics: Algebra, Applied Mathematics, Calculus and Analysis, Discrete Mathematics, Foundations of Mathematics, Geometry, History and Terminology, Number Theory (my personal favorite), Probability and Statistics, Recreational Mathematics, and Topology. No arithmetic. The foundations of mathematics section includes axioms, logic, set theory, and theorem proving. Again, no arithmetic.
If you look at the history of mathematics, different maths were developed to serve the needs of people. All mathematics through calculus were application driven and strongly tied to arithmetical calculations. Early in the 19th century, math went crazy (probably too much math happy juice) and new maths were developed for their own sake. It happens that many of these "pure" maths have applications in today's world, but the applications came later and are only of interest to applied mathematicians (or scientists, engineers, etc.) using them.
Have I put you to sleep yet? LOL If not, here's one of my favorite math quotes - A mathematician is a machine for converting coffee into theorems.