Fixed effects are for removing unobserved heterogeneity BETWEEN different groups in your data.
I disagree with the implication in the accepted response that the decision to use a FE model will depend on whether you want to use "less variation or not". If your dependent variable is affected by unobservable variables that systematically vary across groups in your panel, then the coefficient on any variable that is correlated with this variation will be biased. Unless your X variables have been randomly assigned (and they never will be with observation data), it is usually fairly easy to make the argument for omitted variables bias. You may be able to control for some of the omitted variables with a good list of control variables, but if strong identification is your number 1 goal, even an extensive list of controls can leave room for critical readers to doubt your results. In these cases, it is usually a good idea to use a fixed-effects model.
Clustered standard errors are for accounting for situations where observations WITHIN each group are not i.i.d. (independently and identically distributed).
A classic example is if you have many observations for a panel of firms across time. You can account for firm-level fixed effects, but there still may be some unexplained variation in your dependent variable that is correlated across time. In general, when working with time-series data, it is usually safe to assume temporal serial correlation in the error terms within your groups. These situations are the most obvious use-cases for clustered SEs.
Some illustrative examples:
If you have experimental data where you assign treatments randomly, but make repeated observations for each individual/group over time, you would be justified in omitting fixed effects, but would want to cluster your SEs.
Alternatively, if you have many observations per group for non-experimental data, but each within-group observation can be considered as an i.i.d. draw from their larger group (e.g., you have observations from many schools, but each group is a randomly drawn subset of students from their school), you would want to include fixed effects but would not need clustered SEs.