Great question, but one that might take a book to answer. Let me start with “It depends”.
It depends on many factors. What sport are you training for is a good place to begin. The type of flexibility need by a 5.14 climber is quite far removed from say a runner. The range of motion and the strength needed at the limit of that range of motion for a sport climber will be akin to that of a gymnast or dancer or martial artist. A runner (except for sprinters) typically do not require great range of motion nor do they need near maximum levels of strength at the extremes of that range like the spirt climber. In fact a certain amount of muscular/tendon residual stiffness is an essential quality for runners as it improved their economy due to the elastic return of the energy imparted to those tissues upon impact with the ground.
With this disclaimer and explanation out of the way as perhaps the two extremes of the flexibility/strength spectrum of the kinds of sports we deal with lets look a bit at the middle ground.
Do you feel limited in the function of your sport because of your flexibility? Are you dealing with an injury or some disfunction related to an old injury. Older athletes often need to do some sort of stretching to get their abused bodies back to a functional level. Steve and I are both good examples of this. Each of us has accumulated impressive orthopedic histories. We are both getting older and aging muscles and facia, especially those full of scar tissue need to some special handling.
If you are asking in a more general way, here is more general answer: Stretching muscles while they are under concentric tension as is done when joints are worked through their full range of motion while under load is a great way to develop functional mobility. Believe it or not Olympic style weight lifters have some of the most mobile hips and shoulders of any athlete. Can you almost touch you butt to the floor between your heels while holding twice your body weigh overhead? And then stand up with that weight overhead? Not likely. This is why we mention in our book that developing mobility while doing functional strengthening is probably the best way for most athletes to develop both strength and flexibility.
I hope this helps. This is a huge area about which whole books are written.