Some difference might be there. I wouldn’t say that they are related to the color/appearance of said beasts though.
But as an example, you can tame 1 Cat that is level…13 or something and it could potentially come with different ranks of certain abilities compared to another Cat which is also level 13.
Again, if you want to find all that out, feel free to check the linked website.
In Classic, it’s very different to how you go about choosing pets compared to in BfA. Or I should say, it can be very different.
There’s a lot more uniqueness regarding specific beasts in Classic.
As with my previous example, if you want a pet that attacks very fast, my recommendation would be to try and get Broken Tooth with his 1.0 attack speed. Although he will be VERY sought after by a lot of other hunters out there so…might take you a while to get him once you eventually find yourself in the area.
More on that in the link below
https://www.wow-petopia.com/classic/attackspeed.php
For leveling, I would recommend a Boar.
They come with above average health along with decent armor. Which can be useful for someone who is new with the aspect of handling pets.
They don’t do the highest damage though.
Boars do however come with the active ability Charge, which allows the boar to more easily maintain threat on the enemy over yourself, especially in the early stage of a fight in the open world.
Cats on the other hand, deal a lot of damage while having a decent amount of health. Although their armor is fairly low so, you will notice that a cat takes a bit of damage even in the open world.
Another good suggestion once you reach around level 32-34 is to tame a Gorilla.
They have fairly decent health and armor but also deal good damage.
Again, for more details, feel free to visit the linked website 
Try as many different pets as you can really.
Even if you find that there’s a specific pet type you prefer, don’t hesitate to tame other types as well in order to learn whatever you can from them(if you tame + kill creatures with a pet that has an ability that you don’t know yourself, you will eventually learn how to teach said ability to some other pet types).
As far as I know, the beast will retain the level it had prior to being tamed.
If you tame a pet that is several levels below yourself, then you will have to level that pet up until it catches up to you.
Note that if you’re level 60 in Classic and tame for example a beast at level 15, you will only be able to level it up by killing creatures that are close to your own level(if my memory serves me right).
You cannot fight creatures the same level as that new pet you just tamed, as they wont yield any experience to you. This can make leveling new pets a bit of a hazzard, depending on which one you go after.^^
