Each field is predicated on the idea that an abstraction - something that can only be imagined in a theoretical sense - is a reality that can be described and analyzed. For historians, the abstraction is the past - something that we cannot see and we cannot visit to perform scientific observation; while we can observe the consequences of the past, we cannot go there. For ethnographers, the abstraction is the idea of culture: while we can observe expressions of culture, the concept remains an abstraction.

There can be and there is a great deal of overlap shared by the two disciplines, but there are many historians who study the past with little or no thought to culture. In the same way, many ethnographers do not give consideration (or much consideration) to the past. That said, many historians work a great deal with elements of the past that are cultural; and many ethnographers actively consider the past of the cultures they study.

More importantly, the two disciplines are divided by bibliographies (and sometimes by jargon) that are not shared as much as they should be. That is perhaps the thing that separates the two disciplines the most.

My answer has established a distinction between history and ethnography. The discipline of anthropology is conceived differently in different places. Sometimes - especially in the U.S. - it is taken to include historical archaeology (which appears as a separate department in many European universities). Historical archaeology tends to take the ethnographic concept of culture into the past through the study of "material culture" - the remains of a society that can be collected and analyzed. That approach to the study of culture is by definition reaching into the domain of history, but historical archaeologists often take a different approach: historians originally were tied for the most part to the written record and did not consider material culture. While that has changed for many, historians still tend to be remain largely grounded in the written record.

At the same time, historical archaeologists necessarily deal with the written record - when available - to understand something of the sites that they explore, but they will always be grounded in material culture.

As with ethnography and history, historical archaeology ad history tend to be divided by distinct bibliographies - as well as methodological issues. Given the explosion of secondary literature, there is simply no way that a single scholar can read everything that is available, and so historians tend to read their histories (and their primary sources) and historical archaeologists tend to read the work of other historical archaeologists, reading histories and primary sources in a more narrow fashion to understand specific topics. As a result of this, the questions they ask of the past and the observations they make tend to be very different. I address this problem in my book, Virginia City: Secrets of a Western Past (2012), which is a thirty-year retrospective of dealing with the history (primary and secondary, written sources) and the material cultural remains of the Virginia City National Historic Landmark District. Having worked there as a historian (and as a folklorist!) and having participated in a dozen historical archaeological projects in the district, I am in a position to compare the approaches and to recommend how there can be more overlap. A sample chapter is available here.

An afterword: In the U.S. anthropology departments often include linguistics, demography, and primatology. The relationship of each to the study of the past is different, but I'm not dealing with those overlaps here, since I doubt that is the intention of the posted question.