No difference really. Beech taxonomy can be a little confusing but both copper and purple beeches share a common botanical name: Fagus sylvatica Atropurpurea Group. Within this grouping are various forms and cultivars that will emphasize specific coloring or sizing, typically on the purple end of the scale but usually even very purple beeches will turn a coppery red in fall.

I think for your purposes, you should look for Fagus sylvatica 'Purpurea' (or Fagus sylvatica f. purpurea). This is what most of us know as a "copper" beech.

It may be different in your climate, Ben, but I’ve never seen a copper beech turn ‘bright red‘, at least not the kind of red you’d see in a maple. They go a rusty brown ime.