Warning: Bullshit alert. I've had one class in Cognitive Anthropology and a lot of classes in Cognitive Science. I don't have a degree in either (but I do have bachelors degrees in related fields). This is what I think it's all about. But I'm no expert. IANACA.
Cognitive Anthropology is the application of cognitive science methods to the study of group cognition as mediated by culture. Cognitive Science is the interdisciplinary combination of linguistics, philosophy, psychology, computer science, education, and neuroscience to discover and describe what happens inside a thinking person's brain. Cognitive Anthropology looks not at how a single brain in a vacuum solves a problem, but how a group of brains that can communicate and make tools solve problems. What information in encoded into tools beforehand? What information is shared in real time while the group works on a problem? What hierarchies does a culture create to facilitate group problem solving? How do experts receive their specialized knowledge? What abstractions do cultures use to simplify group problem solving? What similarities are there between the kinds of abstractions a culture uses for one problem and another problem and how are those patterns in abstraction related to other cultural themes. How does culture affect the process of creating solutions to new problems?
Cognitive Science tends to be very structuralist (a feature inherited from Linguistics), describing things in terms of systems, schemas, paradigms, features, inheritance, and composition, always abstracting small patterns to see if they also describe large patterns (kind of like polymorphism in computer science). Cognitive Anthropology adopts this structuralism in it's approach to describing culture (though plenty of other non-cognitive anthropologists are structuralist in their methodology as well).
I have a bachelor's degree in Linguistics and another one in Computer Science so my undergrad course work focused heavily on cognitive science. I took one Cognitive Anthropology course during my last year of school and it bowled me over. It covered much of the ground that I felt was really missing in traditional single-mind-focused cognitive science. If I ever go back to grad school I want to study more Anthropology with a cognitive emphasis.
One of the research ideas I've had in the back of my head for a while has been the idea of doing a Cog.Anth. study of how musical groups communicate in different styles of music. In a jazz band, a rock band, a punk band, a blues band, a symphony orchestra, a chamber group, an experimental improvisation group, etc...what is the hierarchy of members and sounds, how does information flow among the members, what musical cues are there and who gives them, what physical cues are there and who gives them, who is listening to who, who is watching who, how much structure does the composed music itself provide and how much is improvised live, at the composition phase how much structure is provided by the form of the music, how do the design of the instruments constrain the structures of the music and performance? How do musicians use the structures they've learned within their performance realm to model their larger non-musical social sphere.
Two great books to read are: Culture in Mind by Brad Shore and Cognition in the Wild by Edwin Hutchins. These books are both pretty accessible to anyone willing to think hard about stuff (they aren't specialized and jargony) and they are insanely interesting.