Bottom line, it's a preference in-game style. Do you want to rush through the monsters and raid the bosses over and over to try to get the best gear, or do you want to challenge yourself to see if your tactics, reaction to unexpected dangers, and skill can keep your character alive all the way through the end game.
The point is to test the limits of your character. How
much more interesting is it to turn a corner and be
completely surrounded by demons, knowing that if your
unprepared your going to lose everything? Gets your
adrenaline going. You have 1 shot at life. In my
opinion, this almost creates a bond between you and your
HC character. I like the adrenaline rush you get from
those close encounters with death and come out on top.
Things to be aware of about Hardcore Characters:
-Nothing leaves or enters hardcore mode from any other mode.
-You get way more added stress and anxiety that you just don't find in softcore. It would be cool if a dead hardcore character would just revive in softcore mode, but unfortunately they don't.
-Your Banners in hardcore mode will get extra stuff!
-Diablo 3 is a bit harder in hardcore mode than Diablo 2, since you can no longer pre-cast a town portal in the corner of the room to be ready for a quick getaway!
-No real money auction house will be available for hardcore characters.
-The mechanics and drops of hardcore are the same as Softcore. The only difference between softcore and hardcore is the fact that if your character ever dies, it is permanently dead. Everything else is the same.
-There are special achievements for developing hardcore characters.
-I would practice playing softcore mode all the way to level 60 before starting a hardcore character.
-Studying professional hardcore strategy guides for Diablo 3 would greatly help your chance of living. Soul Reaper Legacy's guide provides these guides and more.
-It is possible to Powerlevel Hardcore Characters, but you need to be very cautious about it!
How to deal with a character
This is the game over screen for hardcore characters: