The main differences between being an individual vs a company are the intentions for which a client hires you.
Being 'Designer John Smith' attracts companies hoping to lure you in as a full time employee eventually. This can be good for finding work when you don't decline the possibility of full time work, but can burn bridges when you decide to leave the client.
Being 'John Smith Design Studio' attracts clients looking to offload development to another company in order to meet deadlines and get stuff done.
Being 'Designer John Smith' attracts companies hoping to lure you in as a full time employee eventually. This can be good for finding work when you don't decline the possibility of full time work, but can burn bridges when you decide to leave the client.
Being 'John Smith Design Studio' attracts clients looking to offload development to another company in order to meet deadlines and get stuff done.