Why isn't the APU a standard diesel generator?
Weight.

Piston APUs for trucks are designed for frugal and quiet operation. This one produces 5.2 kW electrical power and weighs 375 lbs.

The APU for the A320 and B737 is a noisy screaming unit that produces 90 kW electrical power, 445 shaft kW. It weighs 375 lbs as well.



The main difference is in the weight of the engine itself. The turbine engine of the A320/B737 APU produces 445 shaft kW and weighs 145 kg. This diesel engine also produces 450 shaft kW and weighs 1542 kg.

Power-to-weight ratios:


The turbine produces 3.1 kW/kg engine weight. 
The diesel engine produces 0.3 kW/kg engine weight, it is a standard industrial machine and not optimised for weight. 
The Rolls Royce Merlin was very much optimised for weight, it used aviation fuel (gasoline) and produced 1.58 kW/kg