Ok, so, if we're going to have a "Space Race," we need to define the racetrack.
There are two really important things to understand about the immediate future (next couple of decades, say) of human spaceflight. One, space flight is really difficult. Two, the main reason space flight is really difficult is that the places we want to go are really far away.
There are only a few destinations worth realistic consideration between now and 2040, and hence part of what I would consider the current Space Race; Near Earth Orbit, the moon and lunar orbit (and the earth/moon Langrangian orbits), near-earth asteroids and comets, and Mars and its moons. That's it, that's as far as humans will possibly get in the next quarter century, if we're very ambitious and very lucky. The one possible addition to this is that if the Dawn spacecraft proves that Ceres is a helluva lot more interesting than currently assumed, it could be prioritized into the mix, but I consider that highly unlikely. The moons of Jupiter and Saturn, tantalizing though they are, are going to have to wait for later generations. Hopefully we will land robot probes there much sooner.
I'm going to talk a bit about linear, point-to-point distances, as a vacuum-packed crow might fly. Spacecraft don't fly in straight lines, a fact I'll be discussing in greater detail at a later point, but the distance ratios for comparison are relatively the same regardless.
Low Earth Orbit, meaning the altitude of the International Space Station, is about 425 km above the earth. That's about the driving distance from Seattle to Spokane, or Chicago to St Louis. To date, only NASA, Roscosmos and China's single Shenzhou 5 mission have successfully put humans here. Only NASA has ever put humans any higher than this.
The moon (and environs) are about 400,000 km away, some 940 times the distance to the ISS. Which is why nobody has been back there in almost 43 years.
At the very closest point in its orbit, Mars is about 100,000,000 km away from earth. That's 250 times the distance to the moon, or about 235,000 times the distance to the ISS. Which, again, is the farthest distance humans have traveled since 1972.
I had intended to create a drawing which accurately depicted these scales, and figured out that I couldn't. The disparity of scale is simply too great. And maybe that illustrates the point as well as any drawing could.