The land of Holodrum is dying and Onox, the General of Darkness, has imprisoned the Oracle of Seasons and is destroying the land. So link must master the changing of the seasons to help him restore Holodrum to prosperity.
The first dungeon in Oracle of Seasons is based on the first dungeon (simply called Level 1, or the "Eagle") in the first Legend of Zelda. They also share the same boss, a fire-breathing dragon called Aquamentus.
Although it does say that its a Game Boy Color game, there are areas of the game that can only be entered if you are playing on a Game Boy Advance. The only confirmed area in Seasons that I can find are the houses with two doors in them. On the GBC only door can be entered, but on the GBA you can enter the second door and find hidden shops.