Let $X$ be a smooth variety and $V, W$ two closed irreducible and reduced subvarieties represented by ideal sheaves $I$ and $J$. Serre defines an intersection multiplicity for an irreducible component $Z$ of $V\cap W$ as $ \mu(Z;V,W)=\sum_{i=0}^\infty (-1)^i \operatorname{length}_{\mathcal{O}_{X,z}} (\operatorname{Tor}^i_{\mathcal{O}_{Z,z}}(\mathcal{O}_{X,z}/I,\mathcal{O}_{X,z}/J)) $ where $z$ is the generic point of $Z$.
The first summand of this sum is $ \operatorname{length}_{\mathcal{O}_{X,z}} (\mathcal{O}_{X,z}/I \otimes_{\mathcal{O}_{X,z}}\mathcal{O}_{X,z}/J) = \operatorname{length}_{\mathcal{O}_{X,z}}(\mathcal{O}_{Z,z}) $ and this is what has a geometric interpretation as the intersection multiplicity for me. Can someone explain to me at a concrete geometric example, why the "naive definition" isn't sufficient? In which often appearing cases is it sufficient?