What are some notable differences between a 20'' trial bike and a BMX bike?
Trial bikes more often have flat handlebars rather than the "tall" bars common on BMX bikes.
Trial bikes are more likely to come with "bash guards" around the chainring.
BMX bikes may come with somewhat higher gears than trial bikes (but not always).
Some trial bikes have no seat or seatpost at all, whereas most or all BMX bikes do have a seat.
Trial bikes have two very strong brakes (often disc brakes), while many BMX bikes have no brakes at all or only one for the rear wheel


Here are two bikes that are reasonable examples of each category (notice the handlebars, the different sized chain ring (i.e. gear transmission ratio), the seat post and the brakes).

The riders are doing seemingly the same trick. But the trial bike stands still like a rock and could stay in this position forever (strong brakes, static equilibrium), the BMX rider is on top of this obstacle only for a few seconds, not needing to use his brake (dynamic equilibrium, notice the fast spinning front wheel):

Trial bike


BMX bike


As for which you should choose, well, I guess it depends on how you intend to ride (and whether you think a bike without a seat is ridiculous!).

Source of both pictures: Wikimedia Commons