The person you pick will tag along with you and Volo, though they don't really do much beyond commentary. The decision actually affects which Legendary Pokemon you'll fight first or second. If you pick Adaman, you'll battle and catch Dialga first. If you pick Irida, you'll battle and catch Palkia first. On the second battle, you will fight the Origin form of the Legendary you did not get.
You are able to get both Pokemon and the items to transform them both into their Origin forms in the post-game so it really doesn't matter which one you pick other than the first battle you use your own Pokeballs for the fight while the second one always gets caught in a special ball that cannot be obtained regularly.
Source: Experience
