As with the previous Nintendogs game, the differences between the different versions is minimal - in fact, the only real difference is the selection of dog breeds to choose from at the start of the game (you can find out which breeds each version has here). In all three versions of the game, there are the same three breeds of cat to choose from - the Standard, your run of the mill feline; a Longhair, with it's characteristic flat face and mega fluff; and the skinny-looking Oriental.
The different versions also get different starting rooms - although you can buy the other ones from the Interiors shop, if there's one you particularly fancy. For Golden Retriever & New Friends, you get the 'Country Home', with it's rustic fireplace and wooden beams; Toy Poodle & New Friends comes with a 'Scandinavian' themed room, with simple creamy wallpaper and wooden floorboards; and French Bulldog & New Friends gives you a 'Modern' room, with grey concrete walls, wood panelled floors and plenty of plants.
Some items - or rather colours of items - are exclusive to each of the versions too, like a Pink Tennis Ball can only be bought in Toy Poodle & New Friends, and a Blue Bell Collar in Golden Retriever & New Friends. You might manage to pick up one of these exclusive toys or accessories if you're lucky enough to StreetPass someone with one set as their present - but other than that, there's no way to get them in your game.