With D&D 5e's Basic Rules released today I think that we should make 5e the default tag that shows up and make dnd-next the synonym (its the other way around currently).
I concur. We should have dnd-5e as the "real" tag, because "Dungeons & Dragons" is what it's being called, and it's the 5th in this particular edition-numbering series.
Then dnd-next should be a synonym of dnd-5e, since consensus seems to be to not have a separate tag for the playtest period rules.
I think there's some value at this point in separating them. The DnD Next questions deal with playtest-specific issues in some cases, so I would be more inclined to have [dnd-5e] and [dnd-5e-playtest] with [dnd-next] linked to the latter. Probably we should also go and look at which questions are still relevant to 5e (based on the Basic Rules) and tag them as both.