ΔE 57.6, loose: different enough that you should pick on purpose. Mahogany Ink is noticeably darker, much more muted, cooler than Transparent Orange.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 58). Mahogany Ink is noticeably darker, much more muted, cooler than Transparent Orange.
The same difference applies the other way round: Transparent Orange is noticeably lighter, much more saturated, 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 Transparent Orange on the colour wheel See Mahogany Ink on the colour wheel
All matches: Transparent Orange equivalents · Mahogany Ink equivalents