Homerun --
The difference is in the finish of the coin -- especially in the fields of the coin (the open areas where there is no design). Proofs were historically produced for collectors and that is still true today, plus a lot of investors now purchase them, as well. The mint uses highly polished dies for striking the planchets (blank disks) and often strike the coins multiple times so that the raised designs (also refered to as the "devices") are fully pressed into the dies for the highest possible relief and the most detail. Modern proofs have mirrored (highly reflective) fields and the raised designs have a "frosted" appearence.
Non-proof coins (a.k.a. "business strikes" because they're intended for commerce rather than for collecting) usually do not have as sharp of a raised design and the fields are not as mirror-like, although there are some coins which may be described as "proof-like" because they possess some of the proof qualities. Proof-like uncirculated coins are usually ones that have been stamped on dies that are still new (before the dies begin to wear and/or get dirty.
Older proof coins were almost always minted in quantities that were a very small percentage of the business strike quantities. In modern times, however, that is not always true--sometimes the amount of proofs issued is greater that the amount of non-proofs. Thus, older proofs are usually worth a lot more than non-proofs, but sometimes with modern proofs, the uncirculated business strike is more rare and worth more, and/or there are instances where modern proofs and non-proofs are worth the same.
Sometimes proofs are identified by where they were minted. Older U.S. proofs used to be minted at the Philadelphia mint only. Today, none are minted in Philly; instead the San Fransico and West Point mints are the only places that produce proofs.
The Red Book -- A Guide Book of United States Coins is a good source of info regarding proofs, their mintage numbers, etc.
Hope this helps.
- Harley-Dog