Sorcerers in the Realms are still associated with the weave though.

And the weave doesn't exist outside the realms.

You don't become a wizard purely through study in the realms. You need to have innate capacity 'the gift', or you are essentially a muggle and can never learn magic, no matter how many books you read. The difference between the sorcerer and wizard is how strong the gift is in them. A sorcerer's gift is strong enough that they begin to manifest magical powers intuitively, whereas most wizards need to be taught spellcasting.