What's the difference between aqueous hydrogen chloride and hydrochloric acid?
There is no difference.

Actually, it would be more accurate to refer to gaseous (and dry) $\ce{HCl}$ as "hydrogen chloride", and to aqueous $\ce{HCl}$ as "hydrochloric acid", but often the terms are used interchangeably.

The difference is due to the fact that, as you might guess, an acid is a proton donor (it means that it tends to give off $\ce{H+}$ in a solution). On the other side, gaseous and dry $\ce{HCl}$ is simply a diatomic molecule. The dissociation in its ions (and the showdown of its acidic properties) only happen in a polar medium (typically water).

In the paper - Zotikov, V.S. Russ J Appl Chem (2009) 82: 1733. https://doi.org/10.1134/S1070427209090389 - you can find a cool example of the difference between a dry hydrogen halide and a wet hydrogen halide (here, the "terrible" HF is involved):


  In  dry  hydrogen fluoride,  carbon  steel  is  subjected to  minor  corrosion  up  to  $\pu{250^\circ C}$.  In  aqueous  solutions the  carbon  steel  is  stable  only  at  the  concentration  of hydrogen fluoride  above  75  wt  %  and  temperature not exceeding $\pu{50^\circ C}$