Apothecary White vs Bright Slate Grey

How close are they?

ΔE 25.3, loose: different enough that you should pick on purpose. Bright Slate Grey is noticeably darker, more saturated than Apothecary White.

Can I use Bright Slate Grey instead of Apothecary White?

Not as a straight swap; see the closer alternatives below. Loose match (ΔE 25). Bright Slate Grey is noticeably darker, more saturated than Apothecary White.

Can I use Apothecary White instead of Bright Slate Grey?

The same difference applies the other way round: Apothecary White is noticeably lighter, more muted.

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.