"Undergraduate thesis" and "Final Paper"
Final Paper is a general decription. It can be the last paper you write for a class or a last paper you write at the end of a school year.
An Undergraduate thesis is specific. It is the paper you write, in whatever your major is, so that you can graduate. You are then awarded a Bachelor's degree.
And a graduate thesis is what you write to earn your Master's degree.
Hope that helps. :-)