I have read that these two British historians are considered two titans of modern historiography, but I want to understand more about why. I am curious about the details of each of their respective theories of history, especially where they disagree. Also, what is their theory of history when compared to non-British Historians like Jacob Burckhardt, or Emile Durkheim? Additionally, which prominent historiographers have come after them, and where do their (Carr and Collingwood's) theories stand in light of Foucault and Derrida?
Someone deeply immersed in Philosophy of History could probably write a total answer to this, and comment on Foucualt, Derrida and Carr. But for the practice of actually doing history, I've found Collingwood pretty useful. Vico had noted that , though history was not something that could be studied in the present ( like we would study a population of beetles), in a way we could understand it better, because we humans actually are making that history, and so are capable of better understanding it than understanding the physical sciences. Collingwood built on that hopeful notion, felt it was important to use available evidence to reconstruct, recreate the past, to actually use our imaginations to do so. Given some of the dry-as-dust compiled histories of his time, it was a good thing to suggest ( Burckhardt I would not put in the dry-as-dust category) . But to do that reconstruction, he said, you have to ask questions - and as what your research shows depends greatly on what questions you ask, you have to think about those questions. Quite a lot of historical writing, then as well as now ( especially popular historical writing), does not think too hard about what questions to ask.
As Collingwood actually spent a good bit of time doing archaeology, he was not a kind of armchair-inhabiting pedant, and his Idea of History is pretty good reading.