Personally I see the advantages of the Cobra Maneuver, but I'm not a fighter-pilot, so what I say probably doesn't matter.
Anyway, one advantage is that performing the Cobra can throw off your pursuer, forcing him to turn away to avoid slamming into you(Since you are essentially slaming on the breaks), and with some luck, line you up for a cheap cannon-kill . (This also applies to missiles if your plane has rear-lockon ability)
The only two downsides is that are the 'to's. The enemy being 'to' close (they can ram you in this case), and/or you being 'to' close to the ground to recover (faceplant into ground = not fun!).