Let $\mathfrak{W}$ be the Lie algebra of vector fields on the line. Via computing extensions between all simple modules in the category $\mathcal{O}$, we give the block decomposition of $\mathcal{O}$, and show that the representation type of each block of $\mathcal{O}$ is wild using the Ext-quiver. To find a connection between $\mathcal{O}$ and the Whittaker category $\Omega_a$, we give an exact functor from $\mathcal{O}$ to $\Omega_a$, which maps simple modules in $\mathcal{O}$ to simple modules in $\Omega_a$ or zero.