I keep hearing my professor try to explain that we can use robust standard errors when we run a regression to confront the issue of heteroskedasticity. However I don't quite understand how telling Stata to use the robust standard errors is different than using regular standard errors. If the regular standard errors have a risk of being a problem wouldn't we always want to use robust standard errors then?
If the assumption of homoskedasticity is truly valid, the simple estimator of the VCE is more efficient than the robust sandwich version. That means it has smaller variance, so your estimates are less uncertain.
Of course, you can always do a heteroskedasticity test first and estimate accordingly.
There's also an interesting point raised by King & Roberts (2014): if your classical and robust standard errors diverge, your model suffers from misspecification that need to be fixed. "Settling" for the misspecified model and just correcting the standard errors will lead to "biased estimators of all but a few quantities of interest."