When does/can one use the likelihood ratio significance test instead of Fisher's exact test or its Pearson $\chi^2$ approximation for comparing two binomial datasets?
Given two binomial datasets (distributions), I'm seeing the LR test being used to compare one distribution against the global (combined) distribution. Usually I apply Fisher's test for comparing one dataset against the other. I realize that LR testing is of the Neyman-Pearson school, which assumes a fully specified alternative model as well as null model. E.g., in the LR test Wikipedia page example, it's being used to compare two binomial datasets (# heads/tails for two coins).
Why not use the $\chi^2$ test to compare the two samples against each other? What are the conceptual differences in these two approaches? When do I use which? And when is it appropriate to compare one sample against not its complement but the global dataset?