"We are finished" vs. "We have finished"

Saying I/we are finished implies that the person in question is in a state of being finished with some task. It is referring to the person and not specifically the task.

I/we have finished refers to the task itself being finished or complete, and perhaps the person has moved on to another task or is waiting for something to occur.

There is a difference in meaning, at least in idiomatic usage.

We have finished indicates that a task is complete. We have no more to do.

We are finished is often used to convey a feeling such as we are doomed or we have lost. There is nothing more that we are able to do.