I have data with an outcome of 0 or 1 (binary) representing success or failure. I also have two comparison groups (Treatment vs. Control). Each subject in the study contributed 2 observations (the treatment is ear drops, so 2 ears). I wanted to model the data and to look for differences between treatment and control. I ran both a generalized linear mixed model (PROC GLIMMIX in SAS) which is a conditional model, and a GEE (PROC GENMOD in SAS), which is marginal. I got very similar estimations of the outcome probabilities in the two groups, and also similar p values. My question is, what is the difference between the marginal and conditional model, in general and in the context of this problem, and how do I know which one to choose and when ?
Either of the models you used are probably fine approaches -- and it's certainly reassuring that the results are similar.
Marginal models are population-average models whereas conditional models are subject-specific. As a result, there are subtle differences in interpretation. For example if you were studying the effect of BMI on blood pressure and you were using marginal model, you would say something like, "a 1 unit increase in BMI is associated with a $Z$-unit average increase in blood pressure" while with a conditional model you would say something like "a 1 unit increase in BMI is associated with a $Z$-unit average increase in blood pressure, holding each random effect for individual constant."
Diggle, Liang, and Zeger (1994) have recommended the use of marginal models when the objective of the study is to make population-based inferences (as is typically the case in epidemiological settings), and mixed models/conditional models when attempting to make inferences about individual responses.
However many others argue that conditional models should always be preferred to marginal models as conditional models can incorporate conditional AND marginal effects. See for example, Lee and Nelder (2004).
I personally prefer to use marginal models, but that's because I'm often concerned with model misspecification (but you may not be). Marginal models are quite robust and less susceptible to biases from misspecification of random effects (obviously). See Heagerty and Kurland (2001).