After a few months of this, you may find the clerk in the bakery recognizing you as you come in, handing you your jelly donut with sprinkles without your having to say a word. Kinda nice when it happens.
Cookies let web servers do the same.
The IP address is required, because it can't send the reply without it. But the IP address doesn't tell it who you are. You usually get a new IP address every time you connect to your ISP, and your old IP address will get recycled and given to another caller after you hang up. Because of that, a web server has no way to recognize you when you come back. You're just another stranger every time you come in, no matter how many times you've been there before, or how recently.