First, apply the within transformation (fixed effects transformation) on a panel data set. Then, apply pooled OLS with dummies for each cross-sectional unit on the same panel data set.
When you compare both regression results you get exactly the same estimates of the coefficients for the exogeneous variables, also the standard errors and the t-values are the same.
Even the unit-specific dummy estimates are equal to the unit-specific unobserved effects you can estimate after applying the within transformation regression. Well, there is one caveat in this respect: You have to add the intercept of the pooled OLS regression to a dummy estimate to get the respective unobserved effects estimate from the within transformation.
The only difference I figured out, is that the within transformation appears to be less efficient in estimating the unit-specific unobserved effects. So, ceteris paribus, the p-values of the unit-specific dummy estimates for the pooled OLS dummy regression are lower than the p-values of the unit-specific unobserved effects from the within transformation (I'm not sure why that happens?).
Hence, performing a pooled OLS regression with unit-specific dummies appears to be supperior to performing a basic within transformation regression. Still, the latter appears to be more often applied. Is there a reason for that? To come back to the title of this post: Is there an advantage of a within transformation over pooled OLS with dummies?
So, as I understood, there appears to be only a computational advantage of the within transformation. Now, the essential question for me is: Why are the p-values of the unit-specific dummy estimates for the pooled OLS dummy regression lower than the p-values of the unit-specific unobserved effects from the within transformation? This really baffles my mind. Because, for small panels (where the computational burden is low), the dummy regression would be advantageous over the within transformation as it yields estimates of the unobserved effects (dummies) with lower p-values?!
EDIT #2: See @ChristophHanck's answer, which solved the mistery of getting different p-values. I incorrectly included an intercept into the pooled OLS dummies regression. After excluding the intercept, the p-values where the same for both models.