Historically, Francis Boas and Claude Lévi-Strauss were among those responsible for the decline of Darwinism within anthropology. Retrospectively, their motivations seem political - often openly so. Boas saw it as his mission to expunge the concept of race from the scientific literature, and replace racial differences with cultural ones.
The document: The Boasian School of anthropology and the Decline of Darwinism in the Social Sciences tells the story of how this happened. It is embarrassing reading - portraying a triumph of ideology over science.
I notice that many of the modern critics of memetics are anthropologists. Their activities generally fit into the historical pattern of Darwin denialism in the social sciences. Even among those who have some sympathies for Darwin, we have documents minimizing the influence of Darwinism - such as How Darwinian is cultural evolution? - Claidière, Scott-Phillips and, Sperber. Their answer seems to be "not very". Boyd and Richerson have a different tack. They accept Darwinism, but take pains to emphasize how cultural evolution is different. My differences remain exaggerated article goes over this strategy. The basics of Darwinism are accepted, but the detailed implications - such as "Hamilton’s inclusive fitness rule" - are rejected.
Why do some anthropologists prefer "cultural variant" to "meme"? My theory is that "meme" is too openly biological - too reminiscent of "gene". Many anthropologists would react to this with a the same forceful immune reaction that they use to reject other biologically-influenced theories. "Cultural variant" is not such "in your face" biology - and so has a better chance of being tolerated by other anthropologists.
One the one hand, it is good to see a few anthropologists finally getting to grips with the application of evolutionary theory to culture. However progress within anthropology is frustratingly slow. The Darwinian revolution is going much more slowly in the cultural realm than it did in the organic realm.
I should add that it isn't just anthropologists who are at fault here. Evolutionists must accept some of the blame. For some reason, many evolutionists prefer to focus their educational efforts on other targets. Instead of educating their fellow scientists, many evolutionists seem to focus their outreach on the unlikely target of creationists. I am sceptical about whether this "gutter outreach" approach is an effective use of resources. I think a "leading from the front" approach would be better. Arguing with creationists makes evolutionists look stupid - in my opinion. It is too much like shooting a fish in a barrel. Evolutionists should spend more time taking on folk closer to their own intellectual size. Resistance to Darwinism within the social sciences is bad news for everyone - but few evolutionists seem interested in fixing the problem.
Part of the problem is specialization. Cultural evolution lies between two stools. Specialization is natural and good - but you have to make sure that you put more workers on the boundaries to make sure that valuable things don't fall down the cracks between the traditional disciplines.
Evolutionists need to clearly explain where their discipline applies to humans. Of course, that should be the job of anthropologists - but they have spent most of the last 155 years fumbling that ball, and still don't seem to have a good grip on it. Thus my equal time for cultural evolution proposal.