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From my understanding, both aqueous hydrogen chloride and hydrochloric acid are written as $\ce{HCl(aq)}$, is there an actual difference between them?

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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 $\pu{50^\circ C}$

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