DMZ and Port Forwarding are two terms often used when talking about internet security, and both of them improve safety. The primary difference is the way they improve security. DMZ is a small portion of the network which is readily available to the internet. Port forwarding is the technique to be able to have still specific functionalities accessible even with the firewall in place.
Port forwarding doesn't add security; however, it does in a sense, since it removes the reason for not putting up a firewall. DMZ permits the admin to implement better security on the rest of the network. Port forwarding is not necessary, and you can still utilize the internet without it. DMZ's are also not as common and are mostly used by massive companies or industries that offer web services.