There's not much difference but you might have network related apps that have been pre-installed during the initial setup process. They will take up some storage space. Also using up battery in the background. You should be able to uninstall them or, at least disable some of them through the apps section in your phone settings.
I don't think there is such a thing as a PAYG phone? I think it depends on the SIM card e.g you could buy a phone with PAYG but you can contact the network and change to pay monthly on the same phone if you wish. Network operators used to lock phones to their own network but that doesn't happen any more so all new phones in the UK are unlocked and will work on any network and any contract. I hope this helps