It can depend a whole lot on how they've designed the sole of the driver head.
I have two 915D2's, in 10.5 and 12. The sole design is such that no matter the adjustment, the club wants to sit with the same face angle. (Square to slightly open.) For me this is a complication, because I want to make face angle changes more than loft changes. I've played these two heads for the past two years, so this is based on tons of experience with them. I know that if I want a slightly different face angle, I have to override the natural way it wants to sit on the ground. Over the two years this has become a bigger and bigger aggravation, and I'm finally deciding that I'm going to move on. Some people scoff at the idea that a golfer can't just square it up by taking the grip correctly. But for me, over time, I do better with a driver that sits on the ground the way I want it to.
My first adjustable was a Nike STR8-Fit Dymo something. I had three lofts of that one, because I found them cheap so why not. On them, the face angle was what moved. So if you wanted the loft change, you'd re-orient the club in your grip.
I have an F7. Cobra claims to have a Smart Sole or Smart Pad that makes it work as I described for the Titleists above. However, it works more like the Nikes. Changing the loft setting makes it sit open or closed, depending.
For me personally it is a big deal. So I think it depends on what method you use to orient your driver in your hands.
Oh, and the lofts DEFINITELY change if the design works like the Titleist. The face angles change in the other examples. And, you can override either one to get the other change, by the way you orient the club in your hands.