To talk about the detonator pyro, I had a discussion with my team's pyro, who is a shotgun pyro, and then bfl satan, who is a flare gun fanatic. Satan is a really good pyro deathmatch, I've seen him kill people so quickly you don't see the flare gun come out. He is also extremely strong with airblasts, he almost never misses them. He can deal with spies in less than a second when he finds them, and i've personally never been able to kill him when i play against him, unless i have the team right beside with two medics, and uber, AND three direct hit soldiers. (although i'm not that good at the game)
My team's pyro, on the other hand isn't that deathmatch focused. He uses the shotgun, and i've more than once turned around to him burning and killing a spy. He plays a very supportive role. Not just for the engie's benefit either, the engie is good enough that the pyro doesn't need to protect his sentry most of the time, mainly because the engie has moved it so the enemy doesn't know where it is.
To get into the abilities of each secondary, the shotgun does more damage, and it doesn't discriminate against if the target is on fire or not. It's a solid hitscan weapon, not much else to say.
The flare gun, it has infinite range, and does 90 to targets on fire. it doesn't have to be reloaded if you switch off. This gives the pyro a hard hitting option for when the axetinguisher isn't in range. It is also possible to spray an enemy pyro with flames and switch to the flare gun with degreaser and shoot them for the crit damage, though this is very difficult.
The detonator can burst in mid air, does mini crits, can remove stickies, can detonator jump a short distance. Unless the medic is down for the other team, or he's retarded, you're not going to do much damage with the detonator over long ranges. On some maps, the detonator isn't viable because the the jump with it is very limited. like on badwater, there are very few locations where you can det jump. To get back on the original map of discussion of this post, steel, you can det jump to E before the bridges are extended to try and put some pressure on it.
Depends on the situation and pyro. I'd also like it to be a rule that if you don't play comp, you shouldn't be allowed to post in this section. Lobbies are not comp.