Flexibility is often confused with the need to stretch and that is then equated with your mobility. So, let's first start by defining flexibility and mobility. Flexibility is your passive (relaxed) range of motion. For example, while standing have someone lift your foot up in front of your body as high as possible. That is your legs flexibility in that position.
On the other hand, mobility is your active range of motion. For example, have someone raise your leg up and now can you lift your leg up and out of that person’s hand? No, then lower the leg and see where you can lift it up to without the assistance of your partner. The height you were able to lift it on your own is your mobility.
There are no scientifically conclusive flexibility training routines. There are a number of factors that impact your flexibility including your age and your stiffness, how tight your muscles are is maybe another way of saying it.
Your flexibility does impact your mobility in so far as it provides a range of motion that while static is a marker of sorts. But, in order be mobile you need to have muscle strength in order to freely move through your range of motion, while also wanting to have a full range of motion to be as mobile as possible.
Everything is also related to your specific activity or sporting preference. An Olympic weightlifter is going to have a different level of flexibility and mobility than an Olympic gymnast, although I train both weightlifters and gymnasts.
Which brings me to why I don't see any separation of flexibility and mobility, and I don't even see any separation by sport because who doesn't want to move as freely as possible, whatever their activity?
In my Gymnastics Strength training program, one of the athletes, Stanley Cecil Bastien, demonstrating the movements is a weightlifter, his picture is the top of this article.
When someone gets injured, it's not always easy to tell whether the injury is the result of some limitation in movement or overuse. If the athlete's ability flexibility and mobility are optimal it does give the athlete more reasons to avoid injuries that can arise from sudden changes in direction, bad form or just plain bad luck in an active situation.
Mobility is end range strength training. Having strength in end ranges will open up an athlete's ability to do more movements and they are less likely to sustain an injury if they have better control of their joints ranges of motion.