I don't believe that there is a universally accepted on definition on the difference, but in my personal opinion, they're different! A design is only the visual qualities of a character design. It's only about how the character looks, nothing more, no written info or name or all. A character is the design + the written information. Who are they, what do they do? Usually, I consider my adopts to be designs until I assign a character to them and start thinking about who they are, at which point I consider 'em characters <:
In short: to me, a design is only the visuals, a character is the design + their info