Here are the three main flaws with the statement:
1) All but the first two paragraphs discuss social aspects of race (i.e. racism), going into detail about racial pseudo-theories and discrimination from colonial times to the Holocaust. It's interesting but it doesn't belong in a statement regarding the scientific aspects of race.
2) The brief paragraph that actually deals with anthropology emphasizes the overlapping across biological populations of single, adaptive traits like skin color and hair form, but it makes no mention whatsoever of skeletal structure and its uses in determining racial affinity in the field of forensic anthropology.
3) The other scientifically oriented paragraph delves into population genetics (not the AAA's field) to argue that large within-group variation renders any between-group variation meaningless. This may have seemed true back in the 70s when the field was still in its infancy, but today it's rejected as Lewontin's fallacy:
In popular articles that play down the genetical differences among human populations, it is often stated that about 85% of the total genetical variation is due to individual differences within populations and only 15% to differences between populations or ethnic groups. It has therefore been proposed that the division of Homo sapiens into these groups is not justified by the genetic data. This conclusion, due to R.C. Lewontin in 1972, is unwarranted because the argument ignores the fact that most of the information that distinguishes populations is hidden in the correlation structure of the data and not simply in the variation of the individual factors.
There is nothing wrong with Lewontin's statistical analysis of variation, only with the belief that it is relevant to classification. It is not true that "racial classification is...of virtually no genetic or taxonomic significance". It is not true, as Nature claimed, that "two random individuals from any one group are almost as different as any two random individuals from the entire world", and it is not true, as the New Scientist claimed, that "two individuals are different because they are individuals, not because they belong to different races" and that "you can't predict someone's race by their genes". Such statements might only be true if all the characters studied were independent, which they are not.
This article could, and perhaps should, have been written soon after 1974. Since then many advances have been made in both gene technology and statistical computing that have facilitated the study of population differences from genetic data. The magisterial book of Cavalli-Sforza, Menozzi and Piazza (13) took the human story up to 1994, and since then many studies have amply confirmed the validity of the approach. Very recent studies (14,15) have treated individuals in the same way that Cavalli-Sforza and Edwards treated populations in 1963, namely by subjecting their genetic information to a cluster analysis thus revealing genetic affinities that have unsurprising geographic, linguistic and cultural parallels. As the authors of the most extensive of these (15) comment, "it was only in the accumulation of small allele-frequency differences across many loci that population structure was identified."
Here's a collection of recent genetic studies that have identified racial clusters: