They're not the same thing — and training only one while ignoring the other is why most people plateau in their movement quality. Flexibility is passive range. Mobility is usable range.
range you control.
You can be flexible without being mobile. A hypermobile person may have more passive range than they can actively control — which is actually a stability and injury risk problem.
Each with a purpose.
Apply consistently.
You can touch your toes on the floor but your squat depth is still limited. Flexibility without motor control isn't mobility — it's just a party trick.
Train the range. Own the range. Then load the range. In that order.