Relationship between electric dipole moment and polarization?
In fact, the first equation is applicable on a microscopic scale  (for an atom or a molecule):
  $\space$ $\vec{p}_\text{ind}=\alpha \vec{E}_\text{local} $ 

While the second one is applicable on a macroscopic scale (for the bulk):
$\space$ $\vec{P}(\omega) \propto \chi^{(1)}(\omega) \vec{E}_\text{ext}(\omega) $ 
So, the polarisability is a microscopic quantity, while the susceptibility is a macroscopic quantity.

You have to notice that the electric field in the first equation is the local field, while it's the external field in the second one.
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