Perhaps and possibly mean the same thing but are used differently in English grammar. "Perhaps" often goes at the beginning of a sentence or before the subject. Where's Paul? Perhaps he went to the store. People don't usually use "possibly" in the sentence above. "Possibly" often appears after modal verbs (may, can, might, will). She couldn't possibly have been at the office. I didn't see her there. You can't use "perhaps" in the sentence above.

Hi, you can say that both have the sense of what may/might happen. Very generally 'perhaps' is more conversational than 'possibly'. Examples: I'll see you tomorrow perhaps at the gym. (There is a chance that I may see you) The reason why there's no reply to the email, is possibly that they are on holiday. (This a sort of explanation)