What's the difference between a Looney Tune and a Merrie Melodie?

Well, after 1943, there's no difference at all but for the different theme songs.  The Warner Brothers cartoon folks arbitrarily designated half their films as "Looney Tunes" and the other half as "Merrie Melodies," using the titles interchangeably.

Before that, however, there was a slight separation.  Looney Tunes tended to be based around recurring characters like Bosko, who was succeeded as their main star by a short-lived hero named Buddy, who was soon supplanted by Porky Pig.  In the meantime, perhaps in emulation of Mr. Disney's "Silly Symphonies," the Merrie Melodies series featured cartoons built primarily around music.  Most of them were structured around a song which served as the title of the short.  (They did not precisely adhere to these distinctions.  The character of Egghead, for instance, turned up in some late-30's Merrie Melodies, breaking the "no recurring character" rule.)

Apart from that, the other main difference was that in 1934, they began to make the Merrie Melodies shorts in color, whereas the Looney Tunes series remained black-and-white until 1943.  From then on, no one knew which was which…or cared.  This 1975 interview with director Friz Freleng makes the point clear…

No, I never knew if a film I was making would be a Looney Tunes or a Merrie Melodie, and what the hell difference would it make, anyway?

That pretty much says it.

By the way: A lot of folks don't know this but the concentric color rings, so familiar to viewers of WB cartoons, have a method to their madness.  The colors were periodically changed to denote different years.  You can read a more detailed explanation, and consult a handy-dandy guide to what colors mean what, over at his website.  Dave also has an excellent gallery of main titles from WB cartoons, and supplied the graphics atop this page.  Thanks, Dave!