The Royal Mail was founded as a public service in 1516 by Charles I and was privatised in 2010.
The Post Office was founded as a public service by Charles II in 1660 and I believed privatised in 1969.
The Royal Mail are the people moving the post around, and the post office is where you go to post something if you're not going to put it in a post box, as well as do a lot of other things like renew your passport, change money into different currencies and you can even bank cash there.