Neuschwanstein is about a 2.5 hour drive from Stuttgart. Although its exterior and setting are magnificent, for me the tourist hordes completely destroy the experience, and the short half hour tour of the interior isn't worth it.
Ludwigsburg ( http://www.ludwigsburg.de ) Palace (in http://www.schloesser-und-gaerten.de )15 minutes away from Stuttgart has the longest (1.5-2.5 hour) and best palace tours of anywhere I've been in Europe, and I've seen quitter a few. There is enough to see and do there to last you several days, and they even have a program where children can dress in period costumes. The extensive gardens include a fairytale one which is fun for the whole family. It is Germany's largest palace in its original condition, that is all buildings and furnishings. There is also a deer park, a small area of century old rides like a carousel, and some birds and animals in the gardens.
For fairytale like castles, there are two excellent ones: Lichtenstein ( http://wwe.schloss-lichtenstein.de ) has a magnificent setting atop a rock on the edge of a cliff, it's small which should appeal to a child, is entered by walking over a drawbridge, and there is a rope's course next door where your children can do things high above the ground and also visitable caves; and Burg Hohenzollern ( http://www.burg-hohenzollern.com ) which sits atop an outliying peak of the Swabian Alb and has true historic significance as it is the family seat of the Hohenzollern family of Romanian and Prussian kings and German emperors. Of these three fairytale-like castlescastles (including Neuschwanstein), I much prefer Lichtenstein which we have frequently revisited.
Stuttgart itself has the Old Palace which is built in the form of a Renaisance Castle and which now holds the Wuerttemberg Museum which has historic items that span a greater range of yime than any other museum of local objects in the world, from mankinds first art and musical instruments from 35,000 yearsd ago through the time when it became a Kingdom in the early 1800s. They also have a special floor of children's exhibitions.
There are also many other nearby castles and palaces that can be visited. The nearby Swabian Alb mountains have by themselves over 400 castles and palaces.
For other things to see and do, have a look at my posts in the thread http://www.tripadvisor.com/ShowTopic-g187275-i116-k6068438-Stuttgart_as_a_base-Germany.html .