Something's drawback or disadvantage is its downside. The only downside you can see to getting a puppy is having to leave it when you go to school — although your parents may see more downsides than that.
When the word downside was coined in the 1680s, it meant "underside" or "underneath." It wasn't until the late 20th century that this word came to mean "negative aspect of something otherwise seen as good." Today, the downside of a puppy isn't its fuzzy belly — it's the hours of training, late-night walks, money spent at the vet, and chewed-up shoes.