Are hosted power counters and advancement tokens the same thing?
Power counters don't (currently) interact with anything other than the card that placed them. A card will define when to place a power counter, and the effects of having them on itself. EG Draco lets you pay to place power counters when it's rezzed, and they boost the strength. The most complicated one currently is Personal Workshop which places power counters on hosted cards, but these cards are inactive while on the workshop, and so the power counters are only referenced by the personal workshop which placed them.

So no, you cannot interchange advancement and power counters.

Advancement and Virus counters are two specific types of counters which a range of cards let you place/remove/move. You can use Trick of Light or Shipment from SanSan to place or move advancement counters. The corp can clear all the virus counters, the runner can play Surge! to place extra virus counters.