The difference is the maneuverability and stall characteristics. The aerobatic planes, of which the Extra 300 is one, are highly maneuverable and have wings that are designed to tip stall with certain inputs to make certain maneuvers better. Tip stalls are great when you want to get a super fast roll or go into a flat spin, but they really suck when you want to land and get one unexpectedly. They are also very neutral in their handling meaning that they do exactly what you tell them and nothing else. Essentially, everything that can be done in the design to make them maneuver faster or have bigger control response has been done. A trainer is the opposite- much slower control response, difficult to stall, and it is designed aerodynamically to fly straight and level. In between the two are the sport planes like Sig 4Stars, Ugly Sticks, and many others that have more forgiving stall characteristics but do have bigger control response than trainers.

To answer your other question- You can use your transmitter and reciever on any plane you might choose to fly. I didn't see that servo listed in Futaba's reference, but anything appropriate for your 300 is too big for a trainer. Before buying a trainer, you might consider visiting the club where you plan to get your training and meet with the instructors. They are likely to have a pet airplane that they are confident in to train with, and sometimes there are connections to used stuff that is still good.