basically, in my experience, the plunge router has a feature where you can change the depth that it extends past the base plate, the trim router is fixed in position in relation to the base plate
All routers have some degree of depth adjustment. The plunge router allows you to "plunge" the bit straight down into the work (provided you have a plunge bit installed) and quickly and fully retract the bit from the cut. Used to make cuts that don't originate at the edge of the work piece and still give you clean cuts that are accurately positioned (using a jig). A trim router is just a motor and base plate with a depth adjustment. They are small and relatively low powered. They are meant for use with a flush trim bit to trim laminate flush with the base material that it is applied to. Between the two are fixed base routers that are real routers with big motors and large bases but without the plunge capability.
What type you need or want depends on what you plan on using it for. A plunge router can do everything that the others can do and would be my first choice if I was only going to have one router.