The difference is that a back tuck is where you jump in the air and basically do a backwards somersault. A back handspring is where you prep, the jump backwards ( like when you start a race that is backstroke) and put your hands on the floor, then keep going backwards for the rest of the rotation until you get back to a standing position.
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There really is no difference, a back tuck is just the correct/technical term
To tuck means to tuck in your legs when you go around instead of just curving your legs. There is a video at http://bestgymnasticsvideos.com/how-to-do-a-back-tuck-flip/that demonstrates how to do the back flip tuck. Watching that should help you see the difference.
A back tuck or "standing" back tuck is done from a non-moving standing position. The tuck pulls the knees toward the chest, as you would do if you curled up into a ball. It helps spin you faster, and clears your feet from the floor. A back flip is done during a tumble pass and the legs are not tucked in.
The tuck motion counteracts the built energy from the flip motion, as you re-extend your legs you are pushing back against the ground, so you negate the momentum.
In a tumble pass, you need to keep that momentum. Tumblers typically learn to "throw" themselves to keep the momentum going for the next move. So a back flip doesn't tuck the legs into the body. The momentum is followed through in a single motion into the next move. If you pay attention to a tumble pass, a tumbler that tucks their backflip will have their feet contact the ground slightly longer after the flip and they seem to slow down. That's because the re-extension of the legs counters that momentum, and they have to use my leg strength and whole body power to throw themselves again to keep going.
So the TL;DR is a back tuck is flipping backward, usually started and ended while standing still, and involves tucking the knees against the body during the flip. A back flip leaves the legs extended and is usually part of a tumble pass to help carry momentum as a "throw" move into the next more advanced skill, and the legs remain extended.