When you start up Brawl's Wi-Fi multiplayer mode, you have two options. They are With Friends and With Anyone. In With Friends, you can only play with friends that you have on your Friend List. Unlike With Anyone, records are actually kept and names will be seen. Opponents will be able to see your icon and comments about yourself before battle. The number of ways you can play is also much more expansive.
In addition to normal Brawls, you can also play co-op Home-Run Contest and Multi-Man Brawl. During normal Brawls, you will also have much more customization options, such as shouting out short, predetermined messages by taunting.