If the cold war space race hadn't happened we'd be on Mars now (using atomic powered ion drive ships), with space stations and probaly a Moon base as well.
Imo it wold be the opposite. Without the cold war space technology would be decades behind. After all it started out as a proxy conflict for prestige. When that incentive isn't there, why bother?
But I agree with your conclusion that the technological development would be different. Yet the general situation of having the same level of (space) technology as now could still happen. Just not 2013 but maybe 1987.