What is the difference between an APU and an ATS?
The ATS is powered by the APU, so you typically need both. It's a bit different for some fighter jets though.

See: How is bleed air used to start a jet engine?

The battery is used to start the APU, the APU then provides electrical and pneumatic power.

The pneumatic power is used for air conditioning, engine starting, pressurizing the hydraulic reservoirs, and other things.

The ATS is a turbine that needs to spin, it is spun using this high pressure air from the APU. When the ATS spins, it spins a shaft connected to the engine, which then spins the engine just enough to compress air and start.