Manned exploration of the Moon has an aspect that is now seldom remembered, but was pretty obvious at the time. Both sides prospected for a permanent Moon presence, soon followed by a populous, self-sufficient Moon colony. Repeating the "New World" scenario.
I don't think such colony would remain non-military (i.e. civilian), but surely expansion into an empty territory is a less destructive (hence more peaceful) goal than destruction of entire life on all its territories (mutual assured destruction). Every penny spent on X means one less penny to spend on Y.
As of now, the Moon race failed to achieve the objective (permanent Moon presence), so there are no practical gains for U.S., or for human race as a whole. These efforts did not paid off... yet. The only things we've earned is some knowledge and some technology that would come handy when we re-attempt space colonization. Well, we can go into how the present use of this bit of technology distributes between "peaceful" and "destructive", but anyway the practical impact is a very small side-effect.
Again, as of now, the effort put into Moon race is nearly useless for our civilization (in practice), but it would be of enormous benefit if we'd finally decide to go and colonize the space.