Hi ElNino, welcome to Hardcore.
Differences between HC and SC:
1) One death allowed
2) If you set the loot option, then someone can collect your gear, but nothing from your inventory or stash. Give this option to trusted friends only or you will probably never see it again.
As for why people kill each other...
Unlike in Softcore, death is our major concern here in Hardcore. People build, play, and level completely different from Softcore. In Hardcore a helping hand is hard to get, unlike in Softcore. People tend to hate rushers and people who can do them absolutely no good. For example: You are a level 47 Barbarian and got rushed all the way to Hell Ancients. More than likely outside of BO, you are worthless to a party and are a liability. Further, people tend to have more knowledge of builds and the game than in Softcore.
Player Killing (PKing) is probably the most fun aspect of Hardcore. People tend to want to kill you only because we can, and it's great fun. It adds to the excitement of the game for us. I've never not killed (PKed) someone and chuckled to myself. I've also lost while PKing and laughed to myself. Why? Because it is a game, and in all games there are winners and losers. Sometimes you win, sometimes you lose, the important thing is to have fun. I joke around with one of our most ardent ANTI-PK forum go'ers here all the time, saying that it is my pleasure to ruin your game for you. That's not the end goal, it's to teach you to become a better player. The more you die, the more you will learn.
Always remember that Hardcore Duelers and PKs (LEGIT ONES MIND YOU) tend to also be some of the best players in the entire game.
When I went from SC to HC long ago, I died a bunch of times. I eventually learned:
1) Resistances in Hell are in fact important
2) Pumping Vita will, in the end, probably save you more times than not
3) When in doubt, go with the safety item/skill
4) There are hack users everywhere out to ruin your game, make some good friends, gain a respected reputation among them, and there is nothing you won't be able to do or have in the game.
5) Legit PKers are to be respected, Illegit PKers are to be run away from.
6) Giving up is the first ticket back to Softcore. Hardcore players tend to be never-say-die kinds of players who truly enjoy Diablo 2 and LOD.
As for first builds, the common suggestion here is always "Go with the build you think you know the best". There is no magic build. For me it was a Necromancer, and it was several Necromancers actually. Softcore gave you alot of practice, now it's time to go LIVE!
Good luck