Simple answer, both.
Flexibility is the ability for a tissue to elongate. Think of your hamstrings when you bend over to tie your shoe. It has to stretch out in order for you to reach your toes. The individual fibers stretch apart, making the muscle longer. Other structures can be stretched as well. Your tendons and ligaments deform and elongate when you stretch along with your muscles. It just happens at a much slower rate. In fact, it is what makes them strong enough to deal with the stress of exercise. They can take a lot of stress put on them because they can elongate to accommodate it.
Everyone does need to stretch. You don’t need to become a master yogi, but if your muscles are able to fully elongate and contract, you have a much better chance and generating more for force and adapting to stress. That means if you are a runner, a weightlifter or just like living your life to the fullest stretching is really important.
Even though stretching is important, you also have to realize why you need to stretch, when and how. Doing too much stretching can actually decrease performance. Stretching when your muscles are cold is not very effective. And stretching in a bad position can damage other structures in your body.
If you find yourself needing to stretch constantly for relief, there are some other things you can do. The biggest is self-myofascial release. The easiest way to do this is with a foam roller. It is a great tool that you can use to compress and relax different areas in your body. If you are looking for a great foam roller, or how to use it check out True Grit Running Company. They have a wide selection of foam rollers and a very knowledgeable staff.
Mobility, on the other hand, is different. It concerns more the movement around a joint. In other words, mobility allows you to perform a selected movement with minimal restrictions. A person that is flexible is not always mobile though. But a person that has great mobility, is usually flexible. To have great mobility you need to have the strength and endurance to control all aspects of the motion and be flexible enough to allow your tissues to elongate as needed.
In my opinion, it is easier to create quicker and longer lasting changes in your mobility. For you runners, this is especially important. You have to have great control of your lower limbs in order to maintain the pace you want and run as effortlessly as possible. If you are doing the Jim Rowland River Valley Run this year, the videos are a great series of mobility drills to do on your off days. They come from a friend and colleague out of Charlotte North Carolina, Dr. Clay Sankey. He is a former collegiate runner who focuses on proper mechanics and functional movement in the athletes he treats. If you want to read more of what he has to offer, follow this link here: TruMotion Therapy