It will do both, but it requires some timing in the program. It has challenges due to the lack of sockets in the w5100 (4). If the server is using all four sockets for inbound requests, you will not be able to connect to a server as a client. No socket available error.
If you want to see how it works, here is a sketch that is a web server and sends email as a client. You would need to change the email send to a html page request.