Being able to twist well is probably most important for safety while driving, but is also critical for most sports like tennis, badminton, golf, and baseball - even running has a twisting component.
Any one-armed pushes or pulls should also involve a twist, as the muscles of the trunk are much larger and stronger than the tiny muscles of the shoulder. Not being able to twist adequately may mean that the shoulder or arm is overworking, potentially creating a shoulder overuse injury.
The best way to treat such a shoulder injury would be to improve the ability to twist. This is an example of why it is useful to not only examine the area that hurts, but also the movements that are painful, as inflexibility in another part of the body may be the root cause of the problem.
Restrictions in one's ability to twist can often come from the organs. For example, the liver is a massive organ that takes up almost half the space in the lower ribcage. If it cannot slide and glide with respect to the ribcage-container or with respect to the other organs that it touches, one's ability to twist will be impacted.
If the contents of the abdominal container are unable to move well, obviously the container won't be able to rotate well either.
Lack of ability to twist the neck is frequently due to nerve adhesions in the soft tissue between the cervical spine and the arms, preventing the nerves from sliding and gliding properly as we move.
We also have very important blood vessels that feed the head that travel up from the ribcage, and if those blood vessels are not able to slide and glide within their soft tissue adequately, neck range-of-motion will be definitely be compromised.
Visceral Manipulation can address these issues, which would make stretching exercises more successful.
Then depending upon your movement goals, we would choose the kind of twist exercises that would help you the most.
Stretching exercises need to be done frequently but gently, as it is important to understand that when we stretch, we are not only stretching muscles, but also blood vessels, nerves and organs.
A "my way or the highway" attitude does not work with stretching. The brain needs to believe the body will not be harmed before it will allow the area that is stretching to let go.