Vanity pets on the beta currently have a pet bar identical to a hunter pet bar. It has the same abilities for "attack", "follow", etc.
Using the attack function doesn't do anything at the moment and only yields an error, but you can tell your companion to move to a certain location!
New Cursor Icon
Certain critters have a new cursor icon as you hover over them. I can't seem to get a screenshot of the cursor, but it looks like a crosshair type icon and replaces the sword cursor icon as long as you're not in combat with the critter. Going into combat with the critter will turn your cursor into the standard attack icon.
So far left and right clicking on these critters with the new crosshair hover icon does nothing and yields a "You do not have permission to perform that function" error. I'm guessing once Pet Battles is fully functional this crosshair will indicate which critters your pet can battle with to either collect it or simply level up from. We'll see!
Currently persisting pets doesn't work on the beta. Using a ground mount or staying grounded while mounted doesn't despawn most vanity pets, however flying away will (of course this doesn't apply to companions that can fly alongside you). Unmounting doesn't seem to auto-summon anymore. This could just be a bug or the function might have been removed. Again, it's still early on so we'll see.
Pet Battles Achievements
There's a new tab in the achievement window for Pet Battles but there aren't any achievements listed there yet. The tab does imply that there might be some specific goals you can work towards while Pet Battling, though. I wonder if any of those will offer pet rewards. :P