Neither one is inherently better than the other, they're very very different, but both enjoyable if they're not diluted (I'll come back to that). Each one rewards a different play style: turn based combat encourages planning ahead and articulating an optimal strategy (or at least close), real time encourages quick thinking and responsive reflexes.
In fact, there's really only one thing you can do to either system that will ruin it for me: dilute it with the other. Nothing annoys me more than a battle system that attempts to satisfy both crowds and winds up ruining itself. The free online flash game Sonny 2 is a good example. The combat is turn based, but they added an element of real time by putting a timer on your turn. If the timer runs out, you wind up doing nothing for that turn. This, of course, leads to a massive disadvantage which stems from nothing but a feeling of being cheated out of something.
In the other direction, the classic Final Fantasy ATB is a real time system ruined by turn based components (it's worth mentioning here that I'm not looking to praise or demean Final Fantasy in any way, shape or form here, just offer a brief analysis of its battle system). I can't count how many times the following scenario occurred for me. Three (or more, but three relevant) characters on the battle field: one almost dead, one at half hp, one at full hp. Healer's turn comes up and I select a heal for the almost dead one. While I was selecting the heal, the enemies turn came up and they chose to punch the almost dead one and end his life. My healer's turn actually comes up and his target is dead causing the AI to select a different target for the heal which inevitably seems to be the one at full hp rather than half.
In short: a turn based system where your action comes up, time stops until you make your choice after which time resumes and that choice executes immediately is fun. A real time battle system where time is constantly flowing and my character responds to my button presses whenever I make them (if only to report an error due to cooldown or mana lacking) is fun. Any combination invoking elements from both systems is annoying at best.