You've probably set your new phone up with your email as IMAP, which is the more modern email system which syncs everything across all devices. If you delete an email, that indicates you do actually want it deleted.
I suspect your old phone was set up as POP, which is the older system. If you really want POP, you can certainly set up your email that way, but you'll have to mark email read in both places, since it's not going to sync anymore.
Matt