The 1187 has a gas compensating system that bleeds off pressure not needed to operate the gas system, allowing light 2-3/4" to heavy 3" shells to be fired in the same barrel. The 1100 must have different barrels (standard and magnum) to do that.
The 1187 has a stainless steel magazine tube. The 1100 has a blued carbon steel tube. Supposedly the 1100 tube will last longer for trap shooting, while the 1187 is corrosion resistant. I have not found any evidence so far of any lifespan difference.
The 1187 and late model 1100's are very similar. The late model 1100 receiver is pretty much an 1187. It uses the 1187 bolt, which has a larger (thicker) extractor. (Early 1100 barrels can be used if their extractor slot is widened a bit on both sides with a file). Many 1100's now use the 1187 magazine cap retention system.
The 1187 uses the standard 1100 action sleeve. 1100 Magnum models use a heavier sleeve. (This applies to the 12ga. The 20ga 1187 uses an even lighter action sleeve.)
As far as target models go (trap, skeet and sporting clays), the 1187 uses non-pressure compensating barrels, same as the 1100. There is very little operational difference between the guns.
Some 1187 barrels intended primarily for heavy loads have slightly altered gas systems, or no gas compsnating system. Turkey, slug, defense and tactical shotguns have no gas compensating system. Field barrels like those commonly used for goose hunting may have slightly reduced gas ports, though they retain the pressure compensating system.
If you're looking for a Rem semi-auto Trap shotgun, and want a fixed choke, the best of these is the 1187 Trap. It has a slightly higher step rib barrel, and is overbored. There are earlier 1100 barrels like this, except as far as I know, all are fixed choke. The earlier 1100 Trap guns have better fit and finish, and the action is smoother. Early 1187 actions are smoother too. The early to mid 1990's are much smoother than those made now.