ΔE 18.7, loose: different enough that you should pick on purpose. Carmine Red is a little darker, more saturated, cooler than Dry Rust Brown.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 19). Carmine Red is a little darker, more saturated, cooler than Dry Rust Brown.
The same difference applies the other way round: Dry Rust Brown is a little lighter, more muted, 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 Dry Rust Brown on the colour wheel See Carmine Red on the colour wheel
All matches: Dry Rust Brown equivalents · Carmine Red equivalents