Seriously, I was arguing this with some friends the other day. I actually got to the point where I had to look the two up I originally thought the two were synonymous.
Hmm, Learn something new every day.
I would have been close.
My answer:
Arithmetic is simple addition, subtraction, multiplication & division of whole numbers, while mathematics is … all-encompassing.
Now let’s go look it up (http://www.m-w.com):
[SPOILER]ARITHMETIC:
a branch of mathematics that deals usually with the nonnegative real numbers including sometimes the transfinite cardinals and with the application of the operations of addition, subtraction, multiplication, and division to them
MATHEMATICS:
the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations
[/SPOILER]
I’m in the same boat, Earthworm Jim.
Arithmetic is part of mathematics.
I would have said arithmetic is simple calculations, especially in your head or on paper, and maths is everything else abstract with symbols in, but I admit that if I’d never done any non-arithmetic maths (say, I forgot everything I learnt in school after I was 11, which isn’t hard for me), there’d be little difference.
I have an easy definition.
If I can do it, it’s arithmetic. If I stare at it blankly for more than 20 seconds, and still can’t figure it out, it’s mathmatics. 
Hmmm, I’d have to take issue with m-w’s saying arithmetic deals with non-negative numbers. True, as taught, it’s usually that way, but since 6 - 3 is equivalent to 6 + (-3), I think it’s kind of a gray area.
Anyway, I think of arithmetic as dealing with the simple manipulation of given numbers, i.e., there are no variables.
Arithmetic is to mathematics as spelling is to writing? (i.e. Thinking that a good mathematician is someone who’s good at subtraction and multiplication is like thinking that a good writer is someone who can spell and punctuate correctly.)
Arithmetic is what a calculator does; mathematics is deciding what to do with the calculator?
“A mathematician is a device for turning coffee into theorems.” -some famous mathematician—Paul Erdos, I think.
Sometimes the word “arithmetic” is used to mean “number theory,” though this usage is somewhat old-fashioned. More commonly, “arithmetic” is one of the “three R’s” kids learn in grade school: “the simple manipulation of given numbers,” as kelly5078 put it, or “ambition, distraction, uglification, and derision,” as Lewis Carroll put it.
Mathematics is a much broader thing, encompassing geometry, symbolic logic, topology, probability theory, analysis, algebra, and so on… some of which have very little to do with arithmetic or calculation or numbers as one usually thinks of them.
Geometry isn’t Arithmetic.
Trigonometry isn’t Arithmetic.
Algebra isn’t Arithmetic either, although there’s a closer relationship (algebra tends to emphasize the rules and structures of the math that governs arithmetic; or, as we put it as kids, “algebra is arithmetic without the actual numbers”). Algebra is to arithmetic what grammar is to composition.
Arithmetic is the boring, tedious part of math. That’s why we teach it first to our children, and drill them on it monotonously year after year. To make them hate math.
To the lay public, arithmetic and mathematics can very well be synonymous because their main use of mathematics is to do arithmetic. So it’s understandable for the comingling of the terms in their minds. But to a mathematician or even someone who has had some training in higher math (even a calculus course), the distinction between arithmetic and mathematics is pretty clear (and has already been posted by others).
There is a reason for that s on the end of mathematics.
Mathematics is made up of many subjects, arithmatic is one, so are geometry, set theory, calculus, and there are several more. Even with very basic level of learning you should know geometry, arithmatic, and basic set theories.
Of course I can. I’m a professional editor.
“Mathematics” has an “e” between the “h” and the “m”.
“Arithmetic” does not.
According to Tom Lehrer:
Counting sheep
When you’re trying to sleep,
Being fair
When there’s something to share,
Being neat
When you’re folding a sheet,
That’s mathematics!
Complete lyrics here.
Mathematics is the general study of relationships within and among sets.
Arithmetic is doing operations on the decimal number system.