The stories are what make each game unique and interesting. Just throwing out a new character and location, and some new vehicles and weapons, doesn't make a new GTA an amazing experience. Games that are not truly meant to be story based will not have a fixed character. You'd have the ability to create any sort of character you'd like.
The storylines also will help familiarize you with the gameplay area. You learn where things are, where to go, how to get there. Also remember those messages you'll see at the top left hand corner while on specific missions. Tutorials to teach you how to do certain things.
The storyline adds to the replay value. Keeps you wanting to play more and more to discover the protagonist's destiny. Some of the missions have you doing unique things that you couldn't do otherwise outside the story. The story just adds so much to the game that without it, the game could just be boring.