AC2 improves on its predecessor in ways that are just inspired and amazing.
The first place that you realize Assassin's Creed 2 is different from the original is in the opening sequence. In the first game, all that Desmond (the real protagonist character, but who is only played occasionally) could do was walk, talk, and (once) pickpocket. Now Desmond runs, jumps, hides, and punches.
Like the first game, Desmond isn't your main character. In AC2, Ezio is the main guy. Here the games depart once again. In the first game, you were told that Altair is an assassin, who has been disgraced in a mission, and must prove himself again. But though Altair is disgruntled and almost quits the order, he remains blindly loyal to his master. None of this is explained, including the question of why Altair became an assassin in the first place. In AC2, Ezio becomes a real character, with a family, a mission, a background, and his own choices. This true storyline and background makes Assassin's Creed 2 a far more engrossing game, and really makes you care about what happens to Ezio.
This background serves to make Assassin's Creed 2 a far more organic game. Though the storyline is slightly more linear, Ezio grows in ways that are not as preordained as Altair. In the first game, as you completed missions, Al-Mualim would give you weapons and upgrades in ways that he saw fit (which were hard scripted by the number of missions you had completed). In AC2, Ezio buys his own weapons and armor, and only occasionally would a linear sequence force him to receive a certain upgrade at a certain time, which makes the game seem far more choice driven.
The game developers also seemed to get the hint that the "Kingdom" area from AC1 was not a big hit. In AC2, each city (3 total, 4 if you count the Villa, which is a sensible thing to do) has a small countryside, and there is no big area with nothing to do to traverse. Instead, the developers gave attention to the cities, implementing an economy with shops, mercenaries, thieves, and prostitutes to spend your money on. A huge improvement came with the addition of a Villa where you basically own your own city and spend money to improve it, then make money off of it.
All in all, AC2 is a huge improvement and a great game for anyone after an amazing storyline, fun game play, and engrossing action.