Why does the C-C bond in ethane have lower polarity than the C-C bond in ethanol?

No, C-C ethane is surrounded completely by hydrogen bonds with a partial positivity. So all around this molecule is partially positive, meaning little change in charge through space, hence low polarization.

For, C-C ethanol there is a single OH alcohol group which is partially negative. This negative alcohol group is contrasted greatly with opposing partially positive hydrogen atoms that are bonded to the other ends of the molecule. The change from negative to positive charges creates a polarity, which is large in comparison to the C-C ethane molecule.

H3C-CH3 vs H3C-CH-OH

In ethane the environment of each carbon is the same, maintaining the complete covalent nature of the bond.

In ethanol, the more electronegative oxygen atom has an electron withdrawing effect pulling the bonding pair of electrons for the C-C bond closer to the carbon bonded to oxygen, giving that carbon a slight negative charge and the other carbon a slight positive charge, making for a slightly polar C-C bond.