Fixed effects vs. cluster-adjusted standard error

Both approaches, using group fixed effects and/or cluster-adjusted standard error, take into account different issues related to clustered (or panel) data. They are distinct approaches, but often you want to use both of them.

First of all, cluster-adjusted standard errors account for within-cluster correlation or heteroscedasticity which the fixed-effects estimator does not take into account unless you are willing to make further assumptions. If you do not want to model the variance-covariance matrix and you suspect that within-cluster correlation is present, I advise using clustered robust standard error because the bias in your SE may be severe (much more problematic than for heteroscedasticity). But you need enough cluster (Angrist and Pischke say 40-50 as a role of thumb). Cluster-adjusted standard errors take into account standard error but leave your point estimates unchanged (standard errors will usually go up)!

Fixed-effects estimation takes into account unobserved time-invariant heterogeneity. This can be good or bad: On the hand, you need fewer assumptions to get consistent estimations. On the other hand, you throw away a lot of variances which might be useful. Some people like Andrew Gelman prefer hierarchical modeling to fixed effects but here opinions differ. Fixed-effects estimation will change both, point and interval estimates (also here standard error will usually be higher).

To sum up: Cluster-robust standard errors are an easy way to account for possible issues related to clustered data if you do not want to bother with modeling inter- and intra-cluster correlation (and there are enough clusters available). Fixed-effects estimation will take use only certain variation, so it depends on your model whether you want to make estimates based on less variation or not. But without further assumptions, fixed-effects estimation will not take care of the problems related to intra-cluster correlation for the variance matrix. Neither will cluster-robust standard error take into account problems related to the use of fixed-effects estimation.