I cannot assess Chess Tempo because I have not used it, but two clear differences between Tactics Trainer and CTS come to mind instantly.
1) Tactics Trainer presents multi-move problems. You might get eight moves right, miss the ninth, score positive points, but still fail the problem. In this sense, it is more realistic and diverse than CTS.
2) The CTS board is too small. Tactics Trainer uses a board that I can see.
TT is very good, though it takes some getting used to. I have a gold membership, which gives me 25 problems a day. That's probably more than enough problems for anyone.
The point of TT is to get you to recognize the best continuation in a particular position in the shortest time possible. The best move is not always a "tactic" in the usual sense.
Chess tempo is pretty good too; the difficulty of the puzzles go up the higher your rating gets.
Chess Tempo is better than CTS -- if you try a move that is also winning but not best it tells you the move was good but look for the best one and doesn't end the problem in failure like CTS does. Also it has endgame problems.
I don't pay for this site so don't know how good the later problems are, this site doesn't have a bad trainer from what I've seen so far, but chess tempo is at least as good and it's free 
Although to be fair I feel like I should mention chess.com advertises multiple benefits of joining, not just for the tactics trainer.
Hey, I'm looking into buying premium membership for some of the features here, but I would like to know what separates Tactics Trainer from free alternatives such as Chess Tempo and Chess Tactics Server. Thanks!