It is important to realize that mobility and flexibility are not the same thing. To me, flexibility is one's ability to move a joint through a range of motion with assistance (passive range of motion), while mobility is one's ability to move a joint through a range of motion under their own control (active range of motion). An easier way to think of this is having someone stretch you out versus doing the movement by yourself. The picture below does an excellent job of showing the difference between the two:

Personally, I believe our goal should be to obtain more mobility (active) versus flexibility (passive) because if you improve your mobility, your flexibility will improve too. However, if your flexibility increases, your mobility might not always increase. This is due to an increased range of motion through passive stretching instead of active stretching. For example, in the picture above, the figure on the right could perform the stretch on the left, but the figure on the left might not be able to do the stretch on the right due to a lack of stability in the muscles. In order for the person on the right to perform that movement they must have a strong enough hip flexor and quadricep to flex the hip and extend the knee. Also, their hamstring must be strong enough to lengthen without straining during knee extension (straightening of leg). And we can’t forget about the most important thing, the core muscles. Without a strong enough core you wouldn’t be able to stabilize the spine and pelvis. This misalignment would not allow you to perform the movement correctly.

As you can see, active stretching is much more challenging and demanding of the body. This demand is what strengthens the muscles and improves mobility. But when you stretch passively you don’t need to activate the same muscle groups. And if the muscles of the body are not challenged enough, eventually they will weaken. When you don’t activate and stress a muscle it becomes weaker and loses its range of motion. But even if a muscle lacks the range of motion to perform a movement the body will still find a way to perform it. A good example of this is when people arch their back to reach overhead or round their back to squat.They lack the mobility in the shoulders and hips to perform the movement so their body finds the extra range of motion in their lower back to perform the movement. Overtime this process takes a toll on the body and eventually leads to  injury.

The human body is simple, it will adapt to whatever stimulus you give it. Thus, if you constantly use assistance during your stretches then you will become better at stretching with assistance. Your muscles will learn to rely on that assistance and once it is gone they will have no idea what to do. I believe that this type of stretching tricks the brain into believing it has a greater range of motion, when in reality it doesn't. The assistance ( person, towel, band, foam roller, etc) used during the stretch increases the stability of the working muscles which is why the person on the left (above) can perform the stretch. But once we removed the assistance, they wouldn’t be able to complete the stretch due a lack of strength in the hips, quads, core and hamstrings. This is the same reason why it isn’t recommended to wear back or ankle braces 24/7. By giving a muscle external support it has no reason to get stronger and will only get weaker. Unfortunately, the majority of people have been stretching like this their whole life. They lack the strength to keep their body stable throughout movement. In order to correct this, you must ditch the passive stretching routine, and instead perform the exercise circuits below: