I am done- is a simple present tense sentence.
It is be used for indicating that you have done a paricular or first task and ready for the next. Like when the teacher writes on the blackboard, you tell her that, I am done with the first part.
I have done- is a simple perfect tense sentence.
It is used after the completion of a task or work, like after completing the whole work from the blackboard you say, I have done.