Before there was Link, there was Yellow Square. Yellow Square had to retrieve the Enchanted Chalice, which had been stolen and hidden away. The Enchanted Chalice was very important to the people of the kingdom, as it had the power to keep beer cold indefinitely.
Like all good video game sequels, Adventure II stick with the basic premise of the original game and just spruces up the graphics and adds a new wrinkle here or there. Once again the Enchanted Chalice has been stolen and once again it's up to Yellow Square to find it. Also once again there are three hungry dragons roaming the countryside.
So you get the idea, right? There's a lot of the same. There are a few power ups now and a minotaur roams the maze, but most of the differences between Adventure and Adventure II are cosmetic.