ΔE 18.2, loose: different enough that you should pick on purpose. Orange Rust is noticeably lighter, more saturated, warmer than Burnt Sienna.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 18). Orange Rust is noticeably lighter, more saturated, warmer than Burnt Sienna.
The same difference applies the other way round: Burnt Sienna is noticeably darker, more muted, 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.
See Burnt Sienna on the colour wheel See Orange Rust on the colour wheel
All matches: Burnt Sienna equivalents · Orange Rust equivalents