The Betaflight option, "Motor PWM Speed Separated from PID Speed" controls whether the flight controller will use synced or un-synced motor output. Synced motor output means that the motor PWM signals are updated at the end of the PID loop. Un-synced motor output means that the PWM signals are generated by a separate hardware unit on the FC, and are independent of the PID loop.
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Some questions I'll address are:
When should you use synced motor output? When should you use un-synced motor output?
What are the benefits and down-sides of synced vs. un-synced motor output?
Is there a benefit to running motor outputs faster than the PID loop?
When might you want to run motor outputs slower than the PID loop?