There is a difference.
This is a demonstrative pronoun. So, using it you, you specify which cat is yours. For example:
-There are many cats in this room. Which one is yours?
-Oh, this is my cat. (pointing at the cat)
But if you enter a room holding your cat, you might say to your guest, "Here is my cat." It sounds like you are introducing it to your guest.
But you could also say "This is my cat" to your guest. Again, it's like you are pointing at the cat drawing the guest's attention to your pet.
"Here is my cat" can mean emphasis on the place:
Where is my cat? Oh, here is my cat. (talking to yourself finding the cat after looking for the animal for some time)