Definition
Flexibility is how far a joint can be moved passively; mobility is how far the client can move it themselves, with control, under load.
A gymnast in a full split has flexibility. A gymnast who can lift into that split without support has mobility. The difference is strength through the range.
It changes the intervention. A client who cannot squat below parallel but can be pushed there passively does not need more stretching, they need loaded work in the bottom position: goblet squats with a pause, deep split squats, tempo work into the end range. A client who cannot get there passively either needs the stretching first.
Test both before deciding. The gap between the passive and active numbers tells you which of the two is missing.