Why is it correct to say "email me", whereas with the word mail we say/write "mail to me"?
The only person I know who says "mail to me" is very much not a native speaker of English. "Write mail to me" is only marginally better.
Mail me that package.
Email me that report.
Please [write|send] me a letter.
Please [write|send] me an email.
I got lots of mail.
I got lots of letters.
I got lots of emails.
Nobody ever sends me any mail.
Nobody ever sends me any email.
Nobody ever writes [to] me.
Nobody ever emails me.
Please mail that card to me.
Please email that picture to me.
As you can see, there are some differences between how the word "mail" is used vs. how the word "email" is used, but that difference isn't that "mail" takes "to" while "email" doesn't. It's more that "email" is used as the electronic equivalent of not just "mail", but also "letter".