Sears is an example of how the introduction of internal competition does not lead to useful innovation - http://www.forbes.com/sites/stevedenning/2013/07/16/do-inter...
(It may lead to increased levels of backstabbing, but that's hardly innovative.)
Humans are a social species, so also have an innate nature to cooperate.
Also, the concept of 'country' is a rather recent invention. See https://en.wikipedia.org/wiki/Nation_state#History_and_origi... for some context. It's certainly possible that kohanz's "hope [that] we someday get to the point where no one really cares what "country" achieved the next space exploration milestone" can be true yet still have competitiveness.
Perhaps space exploration of 100 years from now will be lead by volunteer teams based on, their WoW clan membership. While preposterous, the teams could still be competitive even if they are not organized by nations.