Do humans have enough biological differences to be grouped into races or subspecies?
Firstly, it's not true that you can't tell racial background from DNA. You most certainly can; it's quite possible to give fairly accurate phenotypic reconstruction of the features we choose as racial markers from DNA samples alone and also possible to identify real geographic ancestral populations from suitable markers.

The reason that human races aren't useful is that they're actually only looking at a couple of phenotypic markers and (a) these phenotypes don't map well to underlying genetics and (b) don't usefully model the underlying populations. The big thing that racial typing is based on is skin colour, but skin colour is controlled by only a small number of alleles. On the basis of skin colour you'd think the big division in human diversity is (and I simplify) between white Europeans and black Africans. However, there is vastly more genetic diversity within Africa than there is anywhere else. Two randomly chosen Africans will be, on average, more diverse from each other than two randomly chosen Europeans. What's more Europeans are no more genetically distinct overall from a randomly chosen African than two randomly chosen Africans are from each other.

This makes perfectly decent sense if you consider the deep roots of diversity within Africa (where humans originally evolved) to the more recent separation of Europeans from an African sub-population.

It's also worth noting that the phenotypic markers of race don't actually tell you much about underlying heredity; for example there's a famous photo of twin daughters one of whom is completely fair skinned, the other of whom is completely dark skinned; yet these two are sisters. This is, of course, an extreme example but it should tell you something about the usefulness of skin colour as a real genetic marker.