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Semi-directed, random polymers can be modeled by a simple random walk on $Z^d$ in a random potential -${(\lambda +\beta \omega \left(x\right))}_{x\in Z^d}$, where $\lambda \>0$, $\beta \>0$ and ${\left(\omega \left(x\right)\right)}_{x\in Z^d}$ is a collection of i.i.d., nonnegative random variables. We identify situations where the annealed and quenched costs, that the polymer pays to perform long crossings are different. In these situations we show that the polymer exhibits localization.