To be perfectly honest, I don't think it makes a whole lot of difference. My children are fine with all of our animals. But they're not big animal lovers. I think that the instinct to nurture is either there or it's not. I don't particularly have it with domestic animals - I love my cats because they don't really need me to do anything except put their food out. We tried a dog, but I couldn't stand its neediness, and soon grew to resent it being around all the time. And yet, I donate to the WWF, because I want to help preserve animals and their habitats in the wild.
My kids love the novelty of a new animal in the house, but they soon stop noticing that the animal is around and go back to what they were doing before it arrived. They can take or a leave a domestic animal really. I'm not really interested in forcing them to take responsibility for another creature. I'd much rather concentrate on the things they're genuinely and enthusiastically interested in, and teach them to be responsible in other ways. I don't think an animal can change a person's personality - they're already caring, responsible or lighthearted. An animal may just serve to illustrate those character points, but they're already there.
(I was advised to get cats for my allergies, and they gave me asthma - now I'm stuck with it because my asthma isn't really bad enough to justify getting rid of them.)
On the other hand, for children that struggle with emotional or social development (maybe Asperger's, or other autism), I do believe that a pet can be invaluable. It can give them an outlet for emotions that they find hard to express with people, and can give them a clear focus that they can't find in the wider world that is so bewildering and frightening. I doubt that it always works in this way though. But that's a whole different topic really!