Mobility vs. Flexibility
Mobility practice is a daily requirement, whereas flexibility training is not. Mobility practice provides nutritive and lubricative fluid to wash each joint (which it would not happen without doing so), reversing the aging-process. Flexibility training does not do this.
Dr. Nikolay Amosov, a well-respected Russian surgeon, supported the notion of mobility practice with his "1000 moves to Heaven." Dr. Amosov completely rehabilitated from open heart surgery with mobility practice (including with light weight movements, you find in light Clubbell® training). Furthermore, Dr. Asomov claimed that he reversed his aging process through this mobility practice.
Your ability to move through all 3 dimensions (front/back, top/bottom, right/left - the "6 Degrees of Freedom") is absolutely requisite to the health of your joints - and we are as old as our joints! Injuries and general wear-and-tear cause joint compression (squeezing out "Synovial fluid" - your joint's nutrition and lubrication), create scar tissue (called "adhesions") and calcium deposits ("joint salts") as well as rheumatoid ailments. Your mobility practice decompresses your joints washing them with nutritive and lubricative health while breaking up adhesions and calcium deposits so you can continue to move pain-free for the rest of your life. Flexibility training cannot do this, since it moves against tissue tension in one direction.
Flexible Instability - Mobile Stability
You can consider the primary distinction between mobility practice and flexibility training, your percentage likelihood of injury. In other words, possessing good flexibility without mobility predisposes you to injury! Physiologically, you'll be much safer slightly tight with good mobility, than very loose with inadequate mobility.
I see many injuries in martial arts related to concentration on flexibility and neglect of mobility. Mobility practice improves strength and stability in the extreme ranges of motion of your joints, whereas flexibility training (without mobility) decreases strength and stability.
Mobility - the Anti-Aging Agent
Mobility practice is more important for not only athletic performance, but also for anti-aging. Warrior Wellness is most often used for a morning (afternoon and evening) super-charge, as well as a warm-up for more strenuous activities like Clubbell® swinging. Another important use for mobility practice is as an "active recovery" session or cycle when you don't want to train strenuously.
Flexibility on the other hand regards the elasticity of the tissues. With conventional flexibility training you use static stretching (with the help of gravity, a partner or object leverage to increase length.
Mobility (also called "Dynamic Flexibility" training - though I dislike this term) regards movement (not position) into the extreme range of motion of each joint through voluntary muscular control (and after a healthy foundation, includes light Clubbell® work with inertia, gravity and leverage.)
Unlike you would find in flexibility training, in mobility practice you don't try to hold an extreme position, but rather you pass through it slowly and smoothly without forcing the tissues to deform, but rather by allowing the muscles to relax voluntarily.
When to use Flexibility Training?
If one insists on flexibility training, then I suggest "active contraction release" (a type of PNF type of) stretching. Although very rigorous, it will improve strength. However, since it is so demanding, only perform this type of training at the end of the day, after all other activities conclude. I've used flexibility training after mobility practice to help me strengthen specific weak points in a full range of motion.
Stretching vs. Yoga vs. Body-Flow
Hatha yoga asana (positions or poses), static stretching training and Body-Flow (dynamic mobility training, including courses such as Warrior Wellness and Be Breathed) are three different agendas. The people whom find the most difficulty learning yoga for instance comprise the most flexible; the same is true in Body-Flow. The common Western misunderstanding is that yoga is "just stretching," when in reality it is the opposite; stretching moves against the tension to deform the tissues, whereas yoga asanas breathes into the tension to allow it to relax while simultaneously providing stabilizing strengthening.
I view hatha yoga asana as a means of surrendering tension through breathing and focus to augment mobility practice. As such, I use asana when I encounter a range of motion in my Body-Flow practice which holds considerable blockage and limitation. I move into the asana to release the tension and as a result allow me to amplify my Body-Flow.
It's important to not associate conventional static stretching with hatha yoga asana. For the difference between Body-Flow and Hatha Yoga read: the Body-Flow Answer to the Yoga Riddle.