The only difference is what Pokemon can have those abilities. They both do the same thing.
Source 1 & Source 2
Correction they both do the same thing but there is a difference.
Huge Power, if acquired via Trace, Role Play, Skill Swap, or Entrainment, will only boost your Attack 1.5x. Pure Power will double your Attack regardless of whether you had the ability naturally or not.
That's the only difference.
Source 3- The actual source is the fourth box from the top. And if thats isnt enough heres another Source 4