Flamethrower does "base" flamethrower damage (since it's the default weapon), can airblast frequently, and has 6 seconds of afterburn.
Backburner does more damage, crits from behind, has 6 seconds of afterburn, and now can airblast but much less frequently.
Degreaser can do everything the base flamethrower can PLUS it offers fast weapon switching at the cost of having only 4 seconds of afterburn.
Really, as a pyro, you'd want to choose between backburner or degreaser. There really is no reason to choose the default flamethrower except for "don't have the backburner or degreaser."
Personally, I'm really happy with the unlocks now in comparison to one another. They both are pretty firmly in sidegrade territory. The default flamethrower however could use a tiny uptweak. Maybe an extra 2 seconds of afterburn or airblasts that cost 16 instead of 20 (an extra two blasts per full ammo capacity). Then you'd have three weapons that are all very good sidegrades to one another.
Kep