What's the Difference?
Thirty years ago in the United States pet doors were generally 'small', 'medium', 'large' and 'extra large'. If you had a cat, you generally used a 'small' unless it was a very big cat.
Then we began seeing 'cat flaps' imported from the UK and people began buying them for cats instead of the 'small' sizes. The 'cat flaps' do differ a bit in design. They tend to be much more square, they have rigid flaps and they're generally plastic framed. Often they will include a 4-way lock.
However, if you prefer a flexible flap or a metal frame or would like the flap to be taller than it is wide there is no good reason not to get one of the original 'small' (or larger) sized pet doors. Likewise, if a particular 'cat flap'is a good fit for your dog and has the features you want, get it!
Just don't tell the dog.