What are the advantages of hardcore mode?

The whole point is the increased challenge. And along with that, the associated bragging rights that come with various Hardcore Mode achievements. Hardcore Mode also provides a variety of ways to show off how hardcore you are, such as new banner sigils, a separate ranking ladder, and the name of your hardcore character appears in red, so everyone can see how badass you are.

It's an alternate style of play that was extremely popular in earlier Diablo games. It encourages a very different, much more conservative, playstyle, and by having 'skin in the game' so to speak, the sense of risk, and thus, the (emotional) rewards of victory are substantially heightened.

There is a very different play style in hardcore which may make the game more enjoyable to play. In soft core mode players have no fear of death and can choose to build full damage and no defense. You might play games with your friends and die over and over again while you desperately farm act 3 inferno. In hardcore you have to play at a safer pace. Making sure your built defensively enough to live through everything.