Trap comes across as a more "serious" game, as opposed to Skeet - and not because one has better shooters than the other.
The setup and etiquette of Trap means that you go to the line, shoot, then bust balls. Skeet, the ballbusting is throughout the round. I was at a club, and a Skeeter said, "You're a Trap guy....why don't you talk?" I said, "Why don't you keep score?"
Part of the difference that the OP (and others) have experienced is that different clubs have different foci. We have both Trap and Skeet (normal and wobble), as well as sporting clays, but the Trap field is more active (about a 2:1 ratio in birds thrown in a year). We shoot SCs infrequently.
Why? Well, one guy can run a Trap field (though two is better) quite efficiently, and three rounds (125 birds) per hour is quite reasonable if everyone is ready to go. Skeet, by virtue of having each person step to the station, take their shots, and move, then move to another station, goes more slowly. To me, it seems that "business breeds business".
If you're looking for a more accurate representation of hunting, it's 5-stand or Sporting Clays.
Trap and Skeet are both far more artificial, but not to be discounted.
My kid is a very good Trap shot; he is fair at Skeet (better than me at both). When we had our Sporting Clays Turkey Shoot, we both used a vintage double, and he kicked ass, though the gun-down, take the safety-off, different course was thrown in.
My point? It's ALL good. Try them all, see what you like.
And Bring a Friend!