We show (under standard assumptions) that there are NP optimization problems for which estimation is easier than approximation. Namely, one can estimate the value of the optimal solution within a ratio of $\rho$, but it is difficult to find a solution whose value is within $\rho$ of optimal.
As an important special case, we show that there are linear programming relaxations for which no polynomial time rounding technique matches the integrality gap of the linear program.