Pikmin 3 includes a handful of extra side-missions starring original series protagonists Olimar and Louie, showing what they were up to while the game's main campaign takes place elsewhere. But while these are a nice addition, they're not hugely substantial.
Each mission essentially acts as another time-limited treasure hunt of the type found in the game's existing Challenge mode. Four of these act as a prologue to the events of the game, and each begins with a brief text message from the President, Olimar and Louie's boss last seen in Pikmin 2, ordering the tiny spacefaring duo to go collect more loot.
You then get to see a piece of Pikmin 3 I'd wager a lot of people missed first time around - the funny videos starring Olimar which you could unlock via an extremely laborious collectible hunt that required multiple save files and a website only accessible via the Wii U console's Internet Browser. Or, y'know, you could just go watch them on YouTube.