We start with his default look.
The White German Shepherd is what Hewie actually looks like. This makes him easy to train and control and has a pretty good life bar. He isn't hostile towards Fiona unless you abuse him and treat him wrong, by kicking him or feeding him bad things. I like this look best because he is a beautiful dog. It's the most well-balanced costume.
The German Shepherd is the standard colors that are normally seen when you think of this breed. Browns, blacks, and tans. Hewie is stronger this way, but he is extra difficult to control. He is more likely to attack Fiona this way as well. :C On the upside, his attacks are much more powerful. Complete the game once (your first playthrough counts) and you'll unlock this style.
The Stuffed Toy makes Hewie look like just that. It's cute, but pretty bulky and kinda dumb-looking. It also weakens his attacks. However, it does make him invincible, so if you want to replay the game to unlock more stuff and different endings, you might want to consider this one. Complete the game on Hard Mode to unlock this style. Also, you need to get a Top Breeder or Dog Lover title during one of your previous playthroughs.