1) Overwork the medic!
There's rarely enough health or water on any map to put out everybody who's on fire, which leaves the job to the medic. And when he's putting out people on fire--or even better, trying to put himself out!--he's not supporting attacks, giving soldiers and demomen the HP boost to jump about with impunity, ubercharging friendlies...
2) Flank attack! (especially true for the backburner)
The pyro is slow-moving compared to the other classes who can jump about like quick little bunnies? If you're attacking the enemy head-on as a pyro, you're doing it wrong. Instead, act like a wild-card; find the medic and extinguish his life, counter enemy flanker scouts by setting them on fire (if they're using the sandman, it won't take very long at all to kill them, and afterburn can do it very effectively after a short, half-second burst, particularly if you control any nearby health packs and/or have already offed the medic), give your friends an unanticipated edge in a soldier-vs.-soldier fight...
3) Neutralize enemy attacks! (for the normal flamethrower)
The two soldiers and the demomen are both distinguished by their primary weapons being explosive projectiles. This means that a pyro with an airblast and some skill or a low ping (reasonable enough to assume at competitive levels) can effectively stop incoming rockets, grenades and stickies dead in their tracks, forcing the soldier to shotgun combat and the demoman to melee. It can also break an ubercharge very effectively, if the pyro has plenty of ammo. It might not do much for the scout, but it can hold them in position for a friendly scout or soldier to finish them off, and again, with their low HP, a quick flamethrower burst will force them to break (especially if you're good with the shotgun or flaregun and can hit them while they run).
So why don't we see more pyros in competitive play, if they can turn half of a "normal" competitive team into glorified shotgunners/melee men? Is the stigma of "W+M1" (so called because of the buttons held down by newbie pyros--move forward and attack) that great?