I promise to keep this short and spoilerfree. Assassins Creed II continues the story right off from where the first game ended, that is to say, you wake up in the exact room where Desmond lay down to sleep at the end of the first game and start off in that same room. From then the story takes off with more gameplay in the "real world" and more gameplay in the new setting of the 1400-1500s of Italy.
Assassins Creed II enhances the mechanics that people came to love in the first game. There is more variety in missions (that is more than 8 types of missions), climbing mechanics have improved by feeling more smooth and fluid, the fighting combat on the ground has been improved with a few more skills added to the repertoire, and finally the ability to use tools. Stealth tactics are still used with more options being given to the player for working in with the dynamic crowds (they can respond to change being thrown or their kind being shoved), and lastly during chases enemies are also more responsive with being able to run and climb up to meet you rather immediately.
Additional enhancements include a monetary system, improved environmental graphics, lengthier gameplay (more than 15 hours, excluding sidequests), trophies, strong voice acting. Of these, the monetary feature feels a bit out of place, it really feels like a beta addition. The only feasible aspect of it that I see is using the money found and earned by looting to just throw on the ground to attract a crowd. Paying to be healed by an apothecary seems kind of backwards based on how the first game was with healing by being synchronized.
If you get the game be warned that there are a few weird issues that really should have been avoided after Ubisoft's past experience with Assassin's Creed. There are random and weird graphical issues that continuously crop up from the moment is player (random screen tearing in places, weird framerate hitches (not that it slows the gameplay down but it is noticeable when you are running around), some slowdown when there are lots of characters on the screen, character models look like they are from Assassins Creed 1 (blocky figures, bland/flat textures on random buildings), all of these though pale in comparison to how beautiful the environment looks.
Ultimately, ignoring the character models, and graphical oddities, Assassins Creed II is definitely better than the first game. With trophies, new mechanics to the gameplay, a lengthier and more varied story, and an interesting and engaging story (with interactive cutscenes) and high production values (we should commend the voice actors and the musical staff, they were excellent), this game is the best in the series.
The DLC packs are fantastic and the story is phenomenal.