If I ask someone "have you got my email?", is it wrong? Or should I ask "have you received my email?"
Which one is gramatically correct, and other than this scenario, how do we differentiate these words for other sentences?
Both "have you got" and "have you received" are correct. However, there is a difference in meaning.
I assume your intent is to ask about reception of an email message. With "have you received", this is the only logical meaning.
"Have you got" can also mean "do you have". For some styles of English this is the only meaning; "have got" is equivalent to "have", and is not the present perfect of "get" (for the latter, "have gotten" can be used instead, but this is limited to American English). For many speakers, "have you got my email" would be interpreted as "do you have my email", which typically refers to the email address. To avoid ambiguity, you can use "did you get my email" or, if present perfect is important, "have you received".
Some guides recommend using "have" instead of "have got" in formal writing, but as indicated here, "have got" is very common in informal speech, especially in the UK.