Pairing is the introduction. The two devices exchange a shared key once and remember each other, which is why a headset only asks for a code the first time. Connecting is everything afterwards: the two remembered devices agreeing to open a link right now.
Almost every 'it says paired but no sound' complaint is a connection problem, not a pairing problem. The record of the introduction is still there, but one side refuses to open the link, usually because it is already connected to something else.
Bluetooth devices remember many pairings but connect to very few at once. A headset that connected to a phone in your pocket will ignore the laptop in front of you, and the laptop shows it as paired and unavailable at the same time.
This also explains why 'forget the device' works so often. It deletes the stale introduction and forces a clean one, clearing whatever mismatched state had built up on either side.