Why is it necessary for pilots to have trim controls on fly-by-wire aircraft?
Airbus FBW aircraft have an Autotrim system, so the trim wheel is only there as a backup. In normal law, the trim system is invisible to the pilot. The sidestick pitch inputs control load factor, unlike in a yoke aircraft where it controls elevator deflection. This means that that the farther you move the stick forward and back the faster it pitches, but at center it is a zero load factor, so it stays at its current pitch angle. 

As an example of the difference, if you want to pitch up 10° in a yoke aircraft you pull back on the yoke until it reaches 10° and hold it there. If you let the pressure off it will return to 0° unless you retrim for 10°. Whereas in Airbus normal law, to pitch to 10° you pull on the sidestick until it reaches 10° then let go. The FBW system automatically maintains the new pitch angle. If you don’t do anything else the FBW system will automatically trim the stabilizer to the new pitch angle. There is no feedback as to elevator deflection because it’s not directly controlling the elevator.

Pilot misunderstanding of the Autotrim function was a factor in the crash of XL Airways flight 888T. When the AoA vanes froze, the computer switched to Alternate 2 law, which does not have the Autotrim function. Due to the AoA fault, the autopilot had trimmed the aircraft full nose up. When the AP disconnected and control law degraded it remained there and notified the pilot that he needed to trim manually. The PF did not realize this, so never moved the trim wheel. With the stabilizer in full nose up trim he didn’t have enough elevator authority to prevent a stall and crashed into the ocean.