Brass is a very musical material for instruments (hence, used on brass instruments such as trumpets, trombones, etc.). It is also versatile sound-wise, and is often the most bright-sounding.
I haven't tried or heard a copper shell snare drum yet. But I have the impression that copper snares are slightly warmer and more mellow, compared to other metal snares.
When I want some added projection and more metallic presence sound-wise, I would go for a steel-shell snare.
But then again, not all snare drums are the same. Many factors are involved - shell construction, shell size, shell thickness, bearing edges, drumhead selection, snare wires, hardware, tuning, etc etc...
If I have to choose one, I'll most likely go for a brass-shelled metal snare drum.