ΔE 16.7, loose: different enough that you should pick on purpose. Royal Purple is noticeably darker, warmer than Amethyst Rayne.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 17). Royal Purple is noticeably darker, warmer than Amethyst Rayne.
The same difference applies the other way round: Amethyst Rayne is noticeably lighter, 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 Amethyst Rayne on the colour wheel See Royal Purple on the colour wheel
All matches: Amethyst Rayne equivalents · Royal Purple equivalents