'The letter she has written' focuses on the result, and tells you that she has just finished writing the letter. We imagine that the complete and signed letter is on her desk or in the post box ready to be sent.
'The letter she has been writing' focuses on the action of writing, and suggests that the letter is not yet finished. Perhaps she has written part of the letter and is taking a break.
This is the key difference between these two aspects of the present perfect (simple and continuous).