Mobility, flexibility and tissue quality are commonly confused, but different, components of range of motion.
What follows are my simple definitions of each, an explanation of why each is important and the implications for overall mobility.
Mobility
The range of movement around a joint that you are able to control.
This is what really matters. It makes no difference how long your muscles are if you can't control them when they're at their longest.
Those extremities of movement are where injuries occur. This is usually because your nervous system recognises a threat as you approach the end of the range you can control & will try to protect you from dislocating or otherwise damaging the joint. Sadly, this response is often an over-reaction and manifests as a torn muscle, cramp, muscle tightness or damage to connective tissue. Damage to the actual joint capsule are rare, usually caused by trauma or overuse, rather than a lack of mobility.
Flexibility
This can be simply described as the length of your muscles.
Or rather, the length that your nervous system will allow them to be when you're fairly relaxed. I'm told that if you're under anaesthetic, it's possible for the surgeon to put your knee on your forehead whilst keeping the leg straight.
Like mobility, flexibility is governed by your nervous system. Stretching helps, because it teaches your nervous system to relax, but simply having "long muscles" isn't an injury prevention strategy on its own: too much flexibility can also be a hindrance. Like so many things, more isn't better, better is better.
Tissue Quality
This is how well your tissues move, both muscles individually and in relation to other structures around them.
Training, dehydration, overuse, traumatic injury and a number of other things can cause muscle fibres to "knot" together, stick to surrounding tissue, develop tight bands within the muscle or simply tighten up in order to protect or aid a weak area.
Using foam rollers, lacrosse balls, yoga tune-up balls, sticks, voodoo bands and a range of other tools can help to maintain good tissue quality.
It's also a good idea to book a regular visit to a physio, physical therapist, sports massage therapist or similar bodywork professional. They're often able to identify issues that you haven't spotted & are likely to be able to work deeper into your tissues than you'd be willing to do yourself (deep tissue work can be painful).
Both tissue quality & flexibility have an impact on your overall mobility, so they're worth working on first, but the ultimate goal has to control at the end of your range of movement around every joint.