Are there any differences between the pets other than aesthetics?
Solution 1:
According to this wiki, all pets are equal.
However, different fish produce different properties when fed to your pet.
Solution 2:
Yes, as I came across. There will be times when you get gold or items dropping at a location where you or your pet on foot can't go.
A flying pet can easily go to that place Shift + Click to pick that for you.
Also, when you're on stairs, your ground pet will travel up and down stairs to kill monsters. Whereas a flying pet can fly over stairs (on aggressive mode) & clear the location for you.