"I need to get more flexible hips"
"I need to get more flexible shoulders"
"The biggest limiting factor for golfers is flexibility"
Mobility vs. Flexibility
Everybody has slightly different definitions of these two items.
I view flexibility as more of a passive mobility - or the ability to get moved into a certain range of motion either via a practitioner/coach or by gravity.
As compared to mobility which I view as more active. You are actively trying to move a joint through a certain range of motion.
Something like a SPLIT SQUAT LATERAL FLEXION, while gravity is passively pulling you down into a range of motion, it requires immense spinal control and eccentric force production.
So there's a grey area between the two, but in general:
Flexibility = Passive
Mobility = Active