To say that Half-Life 2 improves on the original is like saying that a Harley is an improvement from the red tricycle you had as a child. It is not just an evolution of the game to a higher form, it is a revolution that sets incredibly high standards for any game that tries to follow it.
The first Half-Life was a fantastic game that revitalized first-person shooters, and showed us that we could expect more from the genre. It had great multiplayer and fantastic weapons. It had excellent graphics for the time, and added abilities like crouching and crouch-jumping to the basic movement set we can home to expect from Quake. The best thing Half-Life, however, was that it had a compelling story.
None of the other FPS games at the time had much of a story. The original Doom probably had the most interesting, and that was simply a space marine who fights demons from Hell following some kind of accident. The original Quake managed to trim this story down to a space marine who fights demons for no particular reason. Quake 2 attempted to add a story, about space marines who fight aliens, but the story was not especially engaging, and wasn’t even explained in-game.
Half-Life, on the other hand, let you know right off the bat that it was different. The game opened with you inside a subway car on your way to work. Through subway announcements and some sparse on-screen text, you managed to fill in the gaps and conclude that your name was Gordon Freeman, and that you worked as a research scientist at a top-secret government facility called Black Mesa. During the introduction, you were free to wander around the subway car and look at things out the window, but you couldn’t do anything other than wait for the story to begin. When the subway car pulled into the station, you were greeted by a guard, and several scientists, who chastise you for being late to work. You proceed to the “test chamber,” where things go horribly wrong, and you (as Gordon) proceed to acquire armor and your first weapon, the crowbar, before you ever encounter an enemy.