ΔE 15.5, loose: different enough that you should pick on purpose. Black Wash is a little darker, more muted than Battle Mud Wash.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 15). Black Wash is a little darker, more muted than Battle Mud Wash.
The same difference applies the other way round: Battle Mud Wash is a little lighter, more saturated.
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 Battle Mud Wash on the colour wheel See Black Wash on the colour wheel
All matches: Battle Mud Wash equivalents · Black Wash equivalents