There seems to be a lot of dumb commentary on race and biology and a lot of equally dumb commentary on race as a social construct, and an odd sense among some people that the two ideas are at odds.
My training is principally on the "race as a social construct" side of the ledger. I've never had any professional social scientist tell me in the context of this discussion that there are no biological differences between groups identified as "races". The point is that racial identification is usually determined by some relatively superficial biological markers, and it is applied quite discretely in most cases to what is really a continuum. Social conventions like "one drop" rules typically make disadvantage relatively inclusive and advantage relatively exclusive when there is an intermediate case on these superficial markers (go figure! big surprise it would work out that way, huh?). The other point is that these social identities come with a boatload of privileges and obligations and that strongly shapes opportunity in a way that is ultimately (of course) largely socially constructed.
The other point about social construction is the naivete of being "post racial" or "ignoring race". It's precisely because race is reified that you can't just declare such a thing. That's not to say we might be "post racial" in some sense some day. We didn't always see race the way we do today and other countries today don't see race the same way we see race, after all. These things evolve. But it's a very real social category today.
None of that says "biology doesn't matter" or "we're all exactly the same except in the collection of physical identifiers". Social scientists that talk about the social construction of race generally - in my experience - don't speak to that because they don't study that. What they'll say is that genetic distributions are smooth distributions and the thing about the social construction of race is that they put discrete boundaries on those distributions. This is more of an issue in places like Brazil than the United States (as Khan points out the distribution is more discrete for blacks/whites in the United States simply because the category "black" is defined so widely... in Brazil it would be very different - you'd have the continuous genetic distribution chopped up much more haphazardly).