ΔE 18.8, loose: different enough that you should pick on purpose. Royal Blue is much more saturated, cooler than Amethyst Rayne.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 19). Royal Blue is much more saturated, cooler than Amethyst Rayne.
The same difference applies the other way round: Amethyst Rayne is much more muted, warmer.
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 Blue on the colour wheel
All matches: Amethyst Rayne equivalents · Royal Blue equivalents