If you are not thinking one-step ahead and you have no hustle, then you probably won't notice much except in the case where it does not downshift. Maybe if you live somewhere without much traffi and with nice straight grid-pattern of roads and no hills you will get the impression that the transmission is more capable than it is.
If you are thinking 1 step ahead (that is to say actually paying attention to what is coming up) then an auto is often two steps behind because it is not doing what it should at the current moment and it is also not prepared for what will be coming up in the next moment. The experience is the feeling of control you get when the car is always doing what you want and not playing catch-up.
So when you want to take a corner without stomping on a brake and exit the corner efficiently, in control and on power then you can easily do that on a manual but an auto transmission. An auto transmission will be fucking around changing gears in some arbitary way because it has no idea what it should do in the next second, only the current second. Not having the correct gearing around corners is not only unsafe in-part because you do not have full control over your entry speed (granted, brakes can get the job done most of the time) but you also do not have control over your exit speed. You are put in a position where you are invited to use your brakes and risk locking up your wheels when things start going wrong (maybe there is sand, gravel or ice involved) and end up in the ditch or into a tree. You might also just cross your fingers and hope ABS to gets you out of trouble.
It's also very unsatisfying to take a corner slow and/or clumsily.