I've got an instance of both Chrome and Firefox open on MacOS right now, and the UI's use exactly the same amount of space. The navigation bar on Firefox is a little thicker than Chrome, but it's made up for with a slightly thinner tab bar (since FF tabs don't have the blank space above to make them look like physical folder tabs).

Not sure if this holds in Linux or Windows.

EDIT: In fact I just noticed that Chrome was outdated. After upgrading it to the latest build, it uses more screen real estate for the UI than FF Quantum's latest build. The difference is even more pronounced if you have bookbar bars turned on.