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Take $X=\mathbb{P}^1\subset \mathbb{P}^2$ and let $Y$ be a single point $y\in X$. Then $j^*$ gives you a trivial bundle corresponding to $\mathrm{k}$, while $j^{-1}$ is isomorphic to the full stalk $\mathcal{O}_{X,y}\simeq \mathrm{k}[x]_{(x)}$ (here $\mathrm{k}$ denotes the base field). This exactly why one considers the pullback functor in algebraic geometry and not the inverse image functor.