Energy Bonus makes your heroes gain more energy for damage taken. This means that they will ultimate earlier, which is most often of importance in the Arenas. The alternative Energy Efficiency lets your heroes keep some energy after they ultimate. This is relevant only in boss fights and raids, really, as heroes rarely get to ultimate twice in Arenas before the fight is decided. Note also that Energy Efficiency has an extremely minor impact on Heroes with an energy depleting ultimate like Hurok, Elador, Zem, etc.
For your main damage dealers and support heroes in boss fights and raids (Sylphi, Sumiko, Rodan, Lingling, Garrick, etc.) you should choose EE as ultimating often is important. For all other heroes, choose EB. This includes heroes that you use in Arena / CoF only, heroes with energy depleting ultimates, and heroes, whose ultimate you do not use at all in boss fights or at most once (like Vernos, Flash, Octo, Goram, etc.).