Lots of good advice from others here already - but let me add some of my own as someone who is EDI-based.

The tram is timetabled to take longer than the bus between the city centre and the airport. For that end-to-end journey I don't think the tram will ever be quicker. Both the tram and the bus can be badly affected by traffic between Haymarket and Princes Street so neither has a real advantage there. The tram's real advantage can come if you are able to begin/end your journey at Haymarket. Between there and the airport the tram doesn't have to deal with roads and so the journey time has almost no variation. The bus, on the other hand, can still suffer with traffic over that section. The timetable says it will be 5 minutes faster than the tram, and often it will be, but when it isn't it can be much, much worse.

Early in the morning and late at night traffic is unlikely to be an issue and I would go with the bus. During the day, I usually get to Haymarket some other way (the #2 bus goes past home and work) and then use the tram. Since the bus & tram stops are next to each other at Haymarket the bus is still an option if the tram is out of sorts in some way - that's happened to me twice. There's a cab rank as well if you are running very short on time.

The tram may also be useful for those looking for cheaper hotel rooms; I see there's an Ibis near Edinburgh Park that I imagine is relatively convenient by tram and probably cheaper than the many options in town. Although the bus passes a Marriot and a Holiday Inn/Hilton (can't remember which) out by the Zoo I doubt either is as cheap.