Spam bots, if there was no check in place to slow them down, would dwarf real people registrations in all systems always. So let's not confuse these two topics - they are different. One part of a system that allows users to register needs to ensure that you have an identifier for a customer and a way to contact that customer, and other techniques try to ensure that you aren't allowing the spammers in the door. Whether you use regex or sending an activation email - neither of those can tell you whether this email address is or is not a spammer.
That's true, but I would posit that you need a registration email anyway (assuming you even care if the email is valid) because even if your regex is perfect there's no way to detect people simply mistyping their email address in a way that is technically valid.