I couldn't have put it better myself. I'd feel bad too, and try to do something nice, but at the end of the day...a dog is a dog. If a cat is lose it could be a stray or a pet and your dog has no way of knowing the difference-both are fun to chase.So sorry That's a tough situation. Your dog was being a dog, though, and simply exhibiting prey drive. That doesn't make her a bad dog, or aggressive- she is just not cat-safe. Sure up your fence, and make sure she isn't out without supervision, and continue with life. She is still the same dog she was yesterday, and doesn't understand that what she did was wrong. I would definitely try to do something kind for the neighbor, a card or gift basket or something, perhaps?
As an aside, and this is my opinion.. their cat was out loose without being fenced in I am guessing.. So while it is your fault the dog got out, it's also partially their fault that the cat was out. My cat is indoor/outdoor, and if she is killed by a loose dog, and she's outside of my fence, that's my fault. Just how I see it- it's one of the risks of having animals outdoors.
I'd be infuriated that they took my dog away, though (she's easily scared of some men and other dogs). I'm not sure what they need to investigate here.