Up until just a few years ago you almost always got better gas mileage with a stick. Sticks are also superior at pulling trailers/dealing with heavy loads, though some of the newest automatics are contenders. Race performance wise sticks are faster off the line, but they can lose some advantage between shifts over an auto, all other things being equal they tend to top out at a higher speed.
Everything I mentioned was solid up until a few years ago. In just the past few years automatics have caught up to sticks in most categories, and are often more fuel efficient now, but all of this in the past few years. If your stick is about five or more years old (unless it's a Saturn with their shitty sticks) you've probably got a better vehicle than the equivalently spec automatic. If it's less than five years old, maybe.
I might actually buy automatic next time. For me and my history that's saying something, but being an owner of said Saturn and noticing newer automatics often have more speeds than the stick eaquivalent, and things like the Beetle getting better gas milage in auto regardless of town or highway I might just make the switch.