What is the difference between arrogance and pride?

Arrogance derives from the act of arrogating. To arrogate is take credit for something, generally when you don't deserve it.

Pride is concern with being good as such. That is, a proud person doesn't just want values. A proud person wants to be good in his attainment of values. Pride is caring about one's own excellence and moral status.

A proud man wants to know he is moral, so he cares about knowing what virtue consists of, and he ensures he's being virtuous. You might say a proud man is a man with a conscience, but not just with the negative side of feeling guilty when he does something wrong. A proud man's conscience makes him feel proud, i.e. happy about himself, when he does the right thing.

Arrogance: "I'm better than you."
Rational pride: "I'm good, maybe even damn good. I don't worry about how I compare to others."

An observer might ask: "Do you care about anyone else?"

One type of person might answer: "All's fair in love and war. May the best man win!"

Another type might answer: "I favor equal protection of individual rights for all, but beyond that, I focus primarily on my own interests, recognizing that I can often benefit greatly by mutual trading with carefully selected others, both materially and non-materially."

Note that the latter type of response doesn't quite lend itself to "sound bites" or slogans as efficiently as less reality-centered responses. Those who prefer to avoid reality probably prefer slogans. Those who genuinely prefer reality may seem to be comparing themselves to others when they select some potential trading partners over others, but they're just doing it for their own personal interests, not to "put down" or conquer others in some kind of "race to the top."

For a good example of each type of person, one can listen to any of the many keynote addresses Steve Jobs delivered while launching new Apple products and contrast that to a speech by Barack Obama.