Robust statistics use modeling techniques that down-weight or eliminate outliers in the observation set. This is an improvement upon the classical approach (Markowitz), which can sometimes be negatively influenced but these outliers.
Robust optimization refers to optimization problems with data uncertainty, i.e. allocation decisions that perform well in a suitable range of realized values for the uncertain parameters.
Bernd Scherer, Deutsche Bank
Portfolio Construction and Risk Budgeting
Olivier Ledoit and Michael Wolf, Credit Suisse