mobility vs stretching

I'm currently doing the just your body challenge and had two days with mobility workouts scheduled last week.

I first did this workout by Nicole:

A couple days later this workout by Kayla came up:

I havn't explored the new content labelled mobility that much yet and was excited that it was included in the challenge. I was confused after Nicole's workout though because it was essentially a stretching workout. I thought mobility is rather about strengthening certain muscles that are important for doing a lot of movements than stretching. So I was expecting something like bodyweight strength but not with conventional strength moves but rather unexpected moves that increase my range of motions. I was also expecting some moves to be challenging because they are engaging neglected muscles. All of this did happen in Kayla's workout but not in Nicole's which was about stetching so not strengthening muscles but elongating them.

So this leads me to my question: what's the difference between mobility and stretching? For those of you who have done some of the mobility workouts: how do you incorporate them in your routine and what effects do they have on your body? I'm not expecting a textbook answer. I am rather interested in your thoughts. Maybe the difference between the two workouts is also due to the different trainers' view of mobility?