Wastegate job
The wastegate sits on the exhaust side. It opens to bypass exhaust around the turbine once the system reaches the target boost level. Without it, the turbo can overspeed and the engine can see unsafe boost.
Blow-off valve job
When the throttle closes under boost, the compressor is still spinning and the pressurised air has nowhere to go. A blow-off valve vents that pressure, reducing compressor surge and keeping the turbo happier between shifts.
Recirculated vs atmospheric
A recirculated valve sends air back into the intake before the turbo. This is common on mass-airflow sensor cars because the ECU has already measured that air. Atmospheric valves vent to open air and can create drivability issues on some systems if not tuned correctly.
Noise is not the main goal
The main goal is mechanical stability. The sound is a side effect. Choose a valve because it suits the engine management and airflow needs, not just because it is loud.