Okay, here's the real difference.

Limit Breaks in Final Fantasy VII, Overdrives in Final Fantasy X, and Weapons Skills in FFXI are the closest to Hyper Combos because they require a gauge to be charged, which then empties upon execution of the move. FFVI's desperation attacks, FFVIII's limit breaks, and FFIX's trance all work differently.

Besides that, the big difference is that Hyper Combos are just that, COMBOS. They cause a fast sequence of hits that causes the combo meter to rise, and a regular combo can be chained into a hyper combo. Only a few limit break type attacks actually cause multiple hits. ALL hyper combos hit multiple times. This is why Zangief's Final Atomic Buster is not a single piledriver, but a series of piledrivers. A good number of limit break type abilities don't even do any damage at all (Aeris definitely comes to mind, as does Eiko's trance), and trance itself is an ability that, for a period of time, transforms your character to become more powerful as opposed to directly launching an attack.

So yeah, that's the big difference.