Other than having its portrait in your little hover mini-profile thingy, does it make any difference at all which pet is the active one?

Some things only affect your active pet. For example, your active pet is the one that gets zapped by either of the pet zappers. During the Morostide even, your active pet is the one that goes into Jaxon's pumpkin patch. I believe there is a random event that gives stat boosts to your active pet. Also, I think if you kick the gym vending machine, it will affect your active pet, but I'm not entirely sure on that one.

I'm sure there are more things that your active pet can do that I either don't know about or am forgetting, so maybe someone else can add a few things. c:

Just to add to what is saying, when you get an Oracle quest, she will give stats to your active pet. So make sure that your active pet is your battle pet when doing her quests. Also, the v2 zapper doesn't just zap your active pet. You get a drop down menu for which pet to zap. The v1 zapper will only zap your active pet though.

- there are also a few games you play with your active pet, like Rock, Paper, Scisorrs or mind reading. Not sure if it makes a difference on the outcome of the game, I haven't tested this. If I remember correctly if you play RPS, and your pet gets tired of playing, you can play more with another pet. But the reward is so small that, other than to earn achievements, I haven't played in eons.