Difference between polarizability and dipole moment?
First, the quick definitions. A dipole moment ($\mu$) implies a permanent separation of charge across a distance. For example, HF has a positive charge near the hydrogen atom and a negative charge near the fluorine:



Instead, polarizability $\alpha$ indicates the degree to which electrons (and thus charge) re-arrange in an applied electric field. Indeed, when a polarizable molecule experiences any type of electrostatics, there is an induced dipole moment:

$ \mu_{induced} = \alpha \vec{E} $

Now this induced dipole moment (from the polarizability) is completely distinct from the permanent dipole moment. For example, benzene is non-polar (no net dipole moment) but highly polarizable. In the HF case, I could find some large applied field $\vec{E}$ that pushes electrons from the F to H.

Here are my mental analogies: 


A dipole moment is like a hill or tilted plane - water (charge) moves "downhill" and it's always there.
Polarizability is like hitting the water with a cannonball. I can drive water from one side of a bathtub to the other!


The two properties can be related, but it's possible to find highly polarizable molecules with no dipole moment - particularly if they are symmetric. Here's a plot with ~80,000 molecules from PQR - note that all combinations occur.