No, Arkham Asylum is not better than Arkham City, because Arkham City has more action. Arkham City is more entertaining, and it has a better premise. The plot lines of Arkham Asylum are a little bit darker and harder to understand. Arkham City is a better read, and it is more fun.
Arkham City is a better videogame than Arkham Asylum for several reasons. First are the graphics--made two years after the original, Arkham City has much better graphics. The gameplay is better and more "free flowing" as its called. The use of Batman's myriad gadgets is awesome and key to the game.