Suppose we are given the task of discriminating, with minimum error, between a set of states $\{|\psi_1\rangle,|\psi_2\rangle,\ldots,|\psi_N\rangle\}$. In other words, we are given an unknown state form this set and our task is to output which of the states we were given.
Let $P_{err}$ be the probability of incorrectly identifying the state. Under what circumstances is this probability minimized by using a projective measurement?
Maybe you should quantify "advantage". To discriminate $\left|0\right>$ from $\left|+\right>$ there is a POVM that will tell either "surely $\left|0\right>$", "surely $\left|+\right>$", or "I don't know". This cannot be done with projective measurement. The "surely $\left|0\right>$" POVM element is proportional to $\left|-\right>\left<-\right|$ and you can probably piece together the rest from there. –  Dan Stahlke Mar 12 '13 at 12:46