What is the difference between login and register?
Registering for something sometimes involves paying money, while signing in or logging in to something usually does not require a payment.
Should I login after registration?
It can be safe to auto login if the user already has an active session as the correct user during the confirmation step. If you think about it, it's not actually "automatically logging them in" but simply keeping them logged in as they was before. During all that time, there was no reason to end the session.