"Own" implies legal ownership. If you "own" something, then it is your property. You can possess something without owning it by stealing it or borrowing it.

"Possess" is a very formal word. You would normally say "have" instead. "Possession" is much more common, and it's easier to see a difference between the "possession" and "ownership".

["Possession is nine-tenths of the law"][1], but it's not legal ownership.

  [1]: https://en.wikipedia.org/wiki/Possession_is_nine-tenths_of_the_law