A linked game features more or less the same story as a non-linked game except that it contains the proper ending to the Oracles plot, so there's really no reason not to go that route.

The password you got from the end of your first game mainly serves as proof needed to begin a linked game. It also transfers certain data such as which animal friend you acquired, but it doesn't affect anything major. What's more important are the additional passwords you get from NPCs as you progress through your linked game. Repeating these codes to the corresponding NPCs in your original game will unlock rewards such as the Master Sword, the Mirror Shield, Bombchus, and the Biggoron Sword.