Matter of politeness. 나중에 보자 is just a casual form and you should not use this to someone you don't know (or don't know well), older than you and of course your teacher or your grandparents . However, 나중에 뵈요 can be wrong, when you talk to someone superior to you. 뵈다 means see or meet superior such as your boss, teacher, grandparents maybe, and very older person than you. But ~요 must not be with that kind of words, so 뵈요 is not proper expression at all. So 봬다 is correct form, though we don't use 봬다 very much at all. If you say to 나중이 뵐까요? to someone you already know but not superior to you, then it can be fine. However we use 나중에 봐요 much more often. Anyway, your comment on Rubby can be rude, 그래, 어쩌면. is only allowed to say to your close friends who are not older than you and you should not say someone you don't know well. Korean people follow very strict rules of honorific, so it is safer to use honorific always.

나중에 뵈요 - вежливая форма 나중에 보자 - не вежливая форма 나중에 봐요 - нейтральная форма значение одинаково.

똑같아요

뵈요x 봬요o 봬요=뵈어요