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.

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 $\mathcal{N}_y\simeq \mathcal{O}_{X,y}\simeq \mathrm{k}[x]_{(x)}$ (here $\mathrm{k}$ denotes the base field and $\mathcal{N}$ is the normal bundle). This is exactly why one considers the pullback functor in algebraic geometry and not the inverse image functor.