How is having a big cat different than having a big dog?

I'd love to have a big cat like the one rich looking couple did, don't recall their names, she had blond hair. And I was watching it, and thinking - how is it different than having a regular house cat? Sure, it's bigger. But some dogs are little, some are big like wolves. Isn't a big dog kinda like a wolf? Are we mainly talking about if an animal attacks, the damage will simply be greater if it's a bigger/stronger animal? Sorry, wish I were bringing this up/asking it better.