Both of the books I reviewed satisfied our basic qualifications for teaching history; I would be quite happy with using either. Each had its strong points; each left some things out.
I can see why Story of the World is so popular right now. It's enjoyable to read, wide in scope, and, of course, there are many activity books and guides to go along with it. (For me, getting activity books is a bad idea–I think of far too many activities as it is.) It's up-to-date, reflecting the more recent archeological discoveries and theories.
On the other hand, I did find A Child's History of the World to be a bit more fun to read. That could just be a personal preference. It has, on average, a bit less detail, which could be a good or a bad thing depending on your perspective. It's more centered on western civilization, only addressing China, the Americas, etc., when they encounter the West. Since in most cases there's little written history to recount before that point, this doesn't bother me.
There is definitely more in A Child's History that I would leave out–especially the first few chapters on cavemen and the attempts at racial classification in ways that are no longer acceptable. (The book is not racist, but “Aryan” just isn't a simple term of ethnic origins anymore.) And, of course, it stops at the Great War.
It also has in its favor the great advantage that we already own it. Supplemented with other books of historical tales and a few books based on more recent archeological discoveries, it will serve us very well. We'll probably get the fourth volume of Story of the World when we get up to modern times.