And yet they mean the same thing.

No, they don't. "Nappy" has connotations of tightly-curled, wiry, and "unmanageable", as opposed to looser curls. Beauty standards have idealized curly hair for centuries.

Have you ever seen hair-product ads from the early part of the century? They often feature a dark-skinned little girl in pigtails (with "nappy" hair) contrasted with a light-skinned girl in smooth ringlets. Both have curly hair but one has "bad" hair and one has "good" hair.

Lots of white people have "curly" hair, too, but it it's rarely tightly-curled enough to qualify as "nappy". That's where the racist element comes in. "Nappy" is effectively black-only.