Diablo II has everything Diablo I missed, and a lot more.
Unlike Diablo I each character has unique skills that can be learned as you level up (by killing monsters). The only way you can use another character's skills is with a charged ring or object.
The story takes place a while after the events of Diablo I, your mission is to travel east in order to find Diablo and his brothers (Mephisto and Baal).
There are four towns (five if you have the expansion), each one completely different from the other and with different characters. In each town you'll have two solve six quests. Most of them are relevant to the story but some of them are filler.
There are houndreds of new objects, which are classified in normal, magic, rare, unique and set.
This game is more enjoyable than Diablo I, you can run and it isn't as dark, which makes it not scary at all, but easier to play. The graphics are OK, but after four years you'd expect something better.
Diablo II was a great game, but it failed in the graphics area and the plot is a bit unoriginal and predictable, but it's also highly addictive and entertaining.