Mice don't naturally become fat, they only do so if put on a high fat diet. And the obese mouse has definitely not been fed on a sensible mouse diet.
Obese mice also can't groom themselves, hence the difference in coat quality between the two mice, not to mention that it clearly looks sicker than the other mouse.
Considering the fat content of yoghurt, it could mean that it is a more healthy option. Or it could also mean that mice don't really like the taste of it very much. Or it could have no effect, but the other diet given to the other mouse is so unhealthy that it looks like there is an effect.