The difference between the types of ice is similar to the difference between graphite and diamond: same atoms, different arrangement and different crystal structures that are thermodynamically favored at different pressures.

Tangentially related: I've heard that if you want to freeze someone with cryogenics, you need to drop the temperature very quickly to far below normal freezing, so that you get a different type of ice crystal. I'm not entirely sure what the difference is (I'm not even sure this is 100% true), but I suspect whatever harm you're avoiding is related to why frozen food doesn't taste as good. If a chemist, biologist, or cryogenist could correct me, I'd love to hear the details.