HP is not "HiPower" or "High Power", it is "Hollow Point", or Dum-dum rounds to those outside the US, and yes, they are legal for non-military use over here. They are designed to expand on impact/entry into the target dumping more of the bullet's energy into it. They're less effective against "armored" enemies because allot of their energy is spent by the bullet splattering and sending bits all over the place outside of the target instead of inside the target.
The "basic/standard" ammo in game is target/ball, basically what people use at to practice with or when a box of HP is costing about as much as a month's rent, and AP is just that, Armor Piercing/Penetrating.
That covered, as has been stated, Hollow point chews up bears, dogs, wolves, deer and other unarmored critters beautifully. AP is slightly more effective than standard ammo against unarmored critters, not enough better to justify the cost of using only it, but is best used against armored or clustered groups of unarmored critters that you can finagle into a line and dump a mag into.