Welcome.
Here, in no particular order are some answers:
Above ground, also called On-Ground is not, in many towns, considered a permanent structure and while they may require a permit, may not be taxed.
Wet is wet when you get in a pool, regardless.
No diving with an on ground pool. Some may be dished out for some extra depth.
In grounds may have a diving area.
On grounds are round, oval, or in some cases rectangles.
In grounds can be any shape and have lots of features not available to the on ground market
On grounds are generally lower cost to buy and feed. This is due to the lack of excavation (some have dug them in though when they rust through or are emptied, may have issue from exterior pressures.
Scott