Why do PlayStation controllers work differently over USB and Bluetooth on PC?

The physical controller is the same, but USB and Bluetooth don't necessarily present the same interface to Windows or to a game. They can use different device identifiers and different formats for input and output reports. A game that recognizes the USB interface isn't automatically equipped to recognize and interpret the Bluetooth one.

That also applies to features beyond buttons and sticks. Things like rumble, adaptive triggers and controller audio need support from the game and the connection path, so basic input working doesn't guarantee every feature will.

There's a difference between Windows detecting a controller and a game supporting it. The game may use an input API that expects an Xbox-style controller, or it may have its own code for recognizing PlayStation controllers. Either way, the developer has to handle the interface the controller exposes over the connection you're using.

Xbox controllers also have different wired and wireless connection paths, but that doesn't mean they'll have the same compatibility gaps in a particular game. It depends on the controller, API and game.

Steam Input or a controller mapping app can help bridge the gap by presenting input in a form the game understands. The tradeoff is that a game may show Xbox prompts, and translating basic input doesn't necessarily preserve every PlayStation-specific feature.

A Bluetooth adapter or its driver can cause connection problems, but that alone doesn't explain why a game accepts the controller over USB and not Bluetooth. If Windows and other software can see it wirelessly, the game's handling of that interface is a more useful place to look.