With examples from P.G. Wodehouse and The Magnificent Seven.
Okay, maybe not. No, the difference is simply that "gray" is how it's spelled in American English, while "grey" is how it's spelled in British English. Like color and colour, traveler and traveller, maneuver and manoeuvre or whatever fiendish jumble the Brits use to spell that one. (Okay, confession, despite my Midwest U.S. upbringing, I do tend to prefer British spellings on an aesthetic level. Especially manoeuvre.)
As this is a spelling variation that, unlike the other examples above, does not fall neatly into the more well-known categories, a useful way to remember it is that if you are in England, you would spell it as "grey," while in America, you would spell it as "gray."
So the satirically British Bertie Wooster gives us this:
I always used to think that publishers had to be devilish intelligent fellows, loaded down with the grey matter; but I've got their number now. All a publisher has to do is to write cheques at intervals, while a lot of deserving and industrious chappies rally round and do the real work. (Wodehouse, P.G., My Man Jeeves)
While American Wild West gunslinger Chris Larabee leans against the saloon bar and says:
"I'm interested in a man who rides a big gray, may be a lefty. Hired some men out of here a while ago." (The Magnificent Seven [TV], "Nemesis")
One might ask, "But then why do we have a famous medical text written by an Englishman entitled Gray's Anatomy and a popular American television show punning off the title with Grey's Anatomy? Shouldn't it be the opposite?"
This standardized spelling is a relatively new thing. The distinction between U.S. "gray" and British "grey" was not established until the 20th century. If you lived in Canada and your surname was "Gray," it's hardly likely you'd go to the trouble of changing it just because the dictionary folks decided your region preferred it the other way.
Additionally, perhaps because of its recency or its prevalence as a last name, or because it is a miscellaneous spelling variation, the division between "gray" and "grey" is not as strict as, say, that between "color" and "colour," particularly in American English. Most American dictionaries list "grey" as a variant spelling of "gray."
For more on the across-the-pond spelling divide, see chiroho's entry on British for Americans.