Yes, they are synonyms, but encerrar steps it up a bit in terms of intensity, as if you were really firmly closing or sealing something.
As per common usage, fechar can and is applied to anything. To close a door, to close a box, to close a bank account, to close a deal, to close your hands, etc. Encerrar is not so much used for the very straightforward act of physically closing something, but more in other contexts where there’s also the idea of ending something, such as closing a bank account, closing a store for the day (it goes beyond closing the door - the whole business is shut down), closing a company, closing a case or closing a session.
Because of how final encerrar sounds, you can say “fechar um negócio” (close a deal) to mean that you just got it, but if you say “encerrar um negócio”, it’s most likely all over. Someone or something that’s locked in a place presumably for a long time can also be “encerrado(a)”. “Encerrado numa jaula” (locked in a cage), “encerrado no quarto” (locked in the bedroom), and so on.