ΔE 33.8, loose: different enough that you should pick on purpose. Mummy White is noticeably darker, much more saturated than Apothecary White.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 34). Mummy White is noticeably darker, much more saturated than Apothecary White.
The same difference applies the other way round: Apothecary White is noticeably lighter, much 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.
See Apothecary White on the colour wheel See Mummy White on the colour wheel
All matches: Apothecary White equivalents · Mummy White equivalents