Who had the lead in the 60's, 70's, or 80's, highly debatable. However by the CURRENT poll ratio is about 1:1. Despite what the current poll ratio might tell, the sad truth is what it is: the USA has officially dropped out of the space race by closing its space program. The USA will now rely on Russian technology and facilities to do in space research.
So who won the space race? as of today, not the USA.
BBC News - Shuttle makes final touchdown to end US space programme
From article: "Nasa will rely on the Russians to ferry its people to and from the International Space Station (ISS)."
These are the unbiased facts.