trials bikes: hopping around on to and off off things usually at very low speeds the seat can get in the way also never unestamate fashion (Which BMX is also has a lot of). Trials bikes always have had small seats but no seat is one step further, a little in terms of praticality for the pro and a little bit of style / fashion. That is my guess as to why trails bikes have no seat. You never see dirt jumping mountain bikes with out a seat.

BMX: more jumps e.t.c I find that you may start of with a level seat but in soon gets tilted back due to the position you are in when riding around and the heavy landings when you bump down on the seat*, due to the shape and size of a BMX 99% of the time you end on on the back of the seat. A seat is alo reguarly needed to lever against when doing many tricks and grabed. Also getting between places and crusing around you still want a seat.

Although both types of riding are not about speed and fittness as much as other types and rely on balence a lot trail and BMX are very diffrent sports.

* You never want anything over tight on a BMX (excepts wheels cranks e.t.c), if it is slightly loose and you crash it will shift, if you have it rock solid it tends to brake