Passive range versus active control
Flexibility and mobility are related but distinct. Flexibility is the passive range of motion available at a joint, how far it can be moved by an outside force such as gravity or a partner pulling a limb. Mobility is the range a person can actively reach and control under their own muscular power.
A useful illustration is lifting one straight leg in front of you: how high someone can hoist it with their hands is flexibility, while how high they can raise it using only the hip muscles is mobility. The two numbers are often quite different.
Why the distinction shapes training
Because they are different qualities, they respond to different work. Stretching can expand passive flexibility, but turning that new range into usable mobility requires actively strengthening and controlling the joint through it, otherwise the extra range stays passive and unusable under load.
For most physical tasks, controllable active range is what actually matters, which is why mobility training combines stretching with strength work at end range rather than relying on passive flexibility alone. Progress within pain-free limits. This is general educational information, not medical advice.