I'll differentiate analyses using model based versus robust standard errors by referring to the latter as "GEEs" which is in fact an exchangeable definition. In addition to Scortchi's fantastic explanation:
GEEs can be "biased" in small samples, i.e. 10-50 subjects: (Lipsitz, Laird, and Harrington, 1990; Emrich and Piedmonte, 1992; Sharples and Breslow, 1992; Lipsitz et al., 1994; Qu, Piedmonte, and Williams, 1994; Gunsolley, Getchell, and
Chinchilli, 1995; Sherman and le Cessie, 1997.) When I say that GEEs are biased what I mean is that the standard error estimate can be either conservative or anticonservative due to small or zero cell counts, depending upon which fitted values exhibit this behavior and how consistent they are with the overall trend of the regression model.
In general, when the parametric model is correctly specified, you still get correct standard error estimates from the model based CIs, but the whole point of using GEE is to accommodate that very big "if". GEEs allow the statistician to merely specify a working probability model for the data, and the parameters (instead of being interpreted in the strictly parametric framework) are considered a type of "sieve" that can generate reproducible values regardless of the underlying, unknown data generating mechanism. This is the heart and soul of semi-parametric analysis, which a GEE is an example of.
GEEs also handle unmeasured sources of covariation in the data, even with specification of an independent correlation matrix. This is because of the use of empirical rather than model based covariance matrix. In Poisson modeling, for instance, you might be interested in fertility rates of salmon sampled from various streams. The ova harvested from female fish might have an underlying Poisson distribution, but genetic variation that comprise of shared heretibility and available resources in specific streams might make fish within those streams more similar than among other streams. The GEE will give correct population standard error estimates as long as the sampling rate is consistent with their population proportion (or is in other ways stratified).