The primary type is used for Revelation Dance when used by a Dual type Pokemon.
When Flying press is used on a Pokemon with Color Change, the opponent becomes a Fighting type, not a Fighting-Flying.
The secondary type of a Pokemon becomes it's primary type after using Burn Up.
In Gen 1, AI opponents can only see your primary type. In the case that Pineapple (thanks Pineapple,) gave, an AI will spam Toxic on the Bulbasaur line, thinking it's super effective, (another flaw of Gen 1 AI,) when in reality it does nothing, because Bulbasaur's second type is Poison.
So, unless your playing in Gen 1, and you are immune to certain moves because of your type, it doesn't really matter.