Persian Violet vs Amethyst Purple

How close are they?

ΔE 15.2, loose: different enough that you should pick on purpose. Amethyst Purple is noticeably darker, warmer than Persian Violet.

Can I use Amethyst Purple instead of Persian Violet?

Not as a straight swap; see the closer alternatives below. Loose match (ΔE 15). Amethyst Purple is noticeably darker, warmer than Persian Violet.

Can I use Persian Violet instead of Amethyst Purple?

The same difference applies the other way round: Persian Violet 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.