hydrogen chloride and hydrochloric acid have the same formula which is HCl. Apparently, Hydrogen chloride is the name of pure HCl. On the other hand, we call it hydrochloric acid if it is in a solution of water. Now, when HCl is in the solution of water, it form ions. We can write the chemical equation as:

`HCl (aq) -> H^(+) (aq) + Cl^(-) (aq)`

*aq = aqueous, in solution

Ionic solutions have the capability to conduct electricity. Therefore, hydrochloric acid (HCl in solution of water) can conduct electricity because it forms ions.
Unless you put hydrogen chloride (pure HCl) in water, it will not conduct electricity.

hope this helps :)