The loss of memory in Alzheimer's simply means you stop remembering stuff. There's no brain fog so far as I can tell. You just - stop remembering.
It has a lot of unexpected practical effects. You can't give someone new stuff, because they won't recognise it as their's, or remember you gave it to them, and they'll think they've lost the old stuff (or someone has stolen it - which is a common delusion.)
You have to be careful about medication and feeding because they won't remember either. It becomes incredibly easy to overdose on meds by accident, or to forget to eat, or to eat too much.
Likewise shopping. When someone's memory is so poor they can't walk around a store and check things off a list, they need professional help.
After a while conversations become incredibly repetitive. In middle stage AS sufferers literally forget what you said at the start of a sentence before you've finished it. They forget they've already asked the same question six times in the last ten minutes - literally.
You can't teach them anything new at all. So if you try to buy various memory aids or anything that needs a bit of thought or learning, they won't find them useful.
Worst of all is the prospect of personality change. Some people become very agitated, abusive, and even violent - which can be a devastating surprise if they showed no signs of that before.
Eventually they stop recognising people or showing much awareness of their surroundings.
My relative actually became more settled and friendlier, which was a very good thing. But there were a couple of support meetings where we are all wondering which way things were going to go.
If they'd gone the other way, it would have been horrible - well, even more horrible - for everyone.
The onset of Alzheimer's itself is recognizable more by 1. your begining to experience such "brain fog" states more and more often (when otherwise well-rested), and 2. your beginning to experience a bit of "brain fog" all the time, such that you at your "best" is no longer as awake/alert/aware as it was previously.