While these differences are not 100% cut and dried, generally a PokeBody is always on and does not have to be activated.
A PokePower usually has to be chosen to activate.
Again, there are a few exceptions to this, but they are rare.
They are both sub-categories of Pokemon Powers, so anything that effects Pokemon Powers will also affect them.
Conversely, if something affects BOTH PokeBodies AND PokePowers, it will also affect Pokemon Powers.