For me, the key difference is the thing about it being open to the public or not. If you are trying to specifically nurture new young talent, then having every sesion open to the public could be too daunting. I would say that a singers club is as others have described above, with members of public able to wander in and enjoy what is on offer. If you need to seclude your selves in order to do something more akin to teaching/mentoring (at least part of the time) then that is more like a singers circle.

Of course, as the group members gain confidence they may be ready to go public perhaps by turning a regular meeting into a Club night.

I was involved with a community drama & writing project that ran a bit like this, many years ago. We had regular drama sessions behind closed doors and one night in the month it was held in a pub's function room as a poetry and performers Club night. That way the new folks got support and coaching, and a chance to try their material out in public when they felt ready. They also had the inspiration of seeing existing performers do their stuff at the club night.