ΔE 28.3, loose: different enough that you should pick on purpose. Gloomy Violet is noticeably darker, much more muted, warmer than Royal Robes.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 28). Gloomy Violet is noticeably darker, much more muted, warmer than Royal Robes.
The same difference applies the other way round: Royal Robes is noticeably lighter, much more saturated, cooler.
ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.
See Royal Robes on the colour wheel See Gloomy Violet on the colour wheel
All matches: Royal Robes equivalents · Gloomy Violet equivalents