Do GEE and GLM estimate the same coefficients? In a GLM, the likelihood equations depend on the assumed distribution only through the mean and the variance. The likelihood equations are
$$\sum_i^n (\frac{\partial \mu_i}{\partial \eta_i}) \frac{y_i - \mu_i}{Var(Y_i)}x_{ij} = 0, \quad (j = 1, ..., p)$$
and in the quasi-likelihood case, we just let $Var(Y_i) = v(\mu_i)$ be some function of the mean. For GEE, the response is extended to be multivariate, with an assumed correlation structure, with the quasi-likelihood equations.
Does this imply that GEE and GLM will have the same parameter (say $\beta$) estimates (population averaged) with the only difference being correct standard errors in the GEE case (Assuming clustered data?)
If the estimated coefficients are not the same, then what is the difference?
 A: It depends on exactly what you mean and what you're assuming.
If you use the independence working correlation, the parameter estimates $\hat\beta$ in  glm and GEE will be identical, with only the standard errors being potentially different
If you use another working correlation, the parameter estimates $\hat\beta$ will not be identical. They will estimate the same underlying parameter $\beta$ if $$E[Y_{it}|{\mathbf{X_{i}=x_i}]}=x_{it}\beta$$ for a linear model (or the equivalent with a link function for a generalised linear model).  That is, they will estimate the same parameter if the marginal model is correctly specified with respect to past and future $x$, not just current $x$.  (see this question).
If it's only true that $$E[Y_{it}|X_{it}=x_{it}]=x_{it}\beta$$ then the underlying parameters will differ (except with a diagonal working correlation matrix).  What happens is that $Y_{it}-\mu_{it}$ is correlated with $X_{is}$ for $s\neq t$, and using a non-diagonal working correlation matrix brings terms like $(Y_{it}-\mu_{it})X_isV^{st}$ into the expectation of the estimating equations, where $V^{st}$ is the $(s,t)$ element of the inverse of the working covariance matrix.
It's also quite possible that even $$E[Y_{it}|X_{it}=x_{it}]=x_{it}\beta$$ isn't true, in which case the parameters and estimates will again depend on the working correlation matrix.  For example, it could be that the difference in $Y$ for a one-unit difference in  $X$ between people and within people might not be the same. The difference between smokers and non-smokers might be larger than that effect of starting or stopping smoking, either because duration matters or because there are other differences between smokers and non-smokers. Different working correlation structures will weight between-person and within-person contrasts differently. This paper talks about examples
