Fetch and push in this context, are not actually what you know them to be. They are two options available when you are setting up your email account. These two options help in sending emails to the client from the server. The major difference between fetch and push is that, fetch option will require the client to be checking the email at regular intervals in case there's a new email, while push option provides the client the opportunity to receive the emails automatically from the server and you would not have to check for emails.
In other words, fetch is initiated by the client, which the server initiates push. Another difference is that, with push, you get your email faster because it will be processed automatically, fetch takes a little bit longer because you will need to check it by yourself. Finally, fetch consumes more power than push