I chose The Family Kitchen, I think mainly because I liked the layout and recipes a bit more.
It's a great cookbook. I've made several things which I had never tried to do before. For me, even though I could cook most things in this book already, it does provide enough new ideas to make it a worthwhile addition to my cooking library.
Although I've seen neither book in real life, I've seen many of her shows on PBS. I think I'd buy the Family Kitchen because the Ital/Amer Kitchen book has so many old warhorses like lasagna, eggplant parmagiana, Bolognese sauce, etc. that one can find in loads of other cookbooks. Given that she's creative and one of my favorites, both books are probably great, but for the above reason, I'd lean toward the newest one.
Lidia could not write a book not worth having. Her insights about food and cooking in general far exceed the importance of her recipes, which is not to denigrate the importance of her recipes...
Which can just about be said for the work of any truly great cooking writer.