Why does Romeo refer to Paris as a "youth" and himself as a "desperate man" when Paris is older than Romeo in Romeo and Juliet?This is in Act 5...

At this point, Romeo has come to the Capulet tomb to see Juliet's body and to commit suicide. He has bought a powerful potion from an apothecary which he will use for this purpose. Romeo is utterly distraught and has decided that he cannot live without his beloved Juliet, so much so that he has even warned Bathasar to meddle in his plans, telling him that:

"But if thou, jealous, dost return to pry
In what I further shall intend to do,
By heaven, I will tear thee joint by joint
And strew this hungry churchyard with thy limbs:"

Romeo is clearly a "desperate man", in this instance and this is exactly what he tells Paris who confronts him as he opens the tomb. Paris believes that Romeo is about to desecrate the tomb and wants to arrest him. He tells Romeo:

"Stop thy unhallow'd toil, vile Montague!
Can vengeance be pursued further than death?
Condemned villain, I do apprehend thee:
Obey, and go with me; for thou must die."

Romeo's retort clearly shows that he will stop at nothing to fulfill his purpose. He warns Paris, and it is clear from his language that he does not want to fight him.

"I must indeed; and therefore came I hither.
Good gentle youth, tempt not a desperate man;
Fly hence, and leave me: think upon these gone;
Let them affright thee. I beseech thee, youth,
Put not another sin upon my head,
By urging me to fury:'

By addressing Paris as a gentle youth, Romeo is attempting to persuade him to leave. The reference also indicates Romeo's true nature: he is indeed, also gentle driven to desperate means. Note how the terms he uses are soft and not as harsh as those he used with Balthasar. The implication is also that Paris is innocent of any wrongdoing and is not battle-hardened or ruthless and Romeo does not wish to hurt him and commit 'another sin'.

In contrast, Romeo is a 'desperate man', intent on doing what he came for. Although he is younger than Paris, the events of the past few days have made him a man: he killed Tybalt, married Juliet, was banished and has a death-sentence hanging over his head if he should return. Paris has not experienced such desperate circumstances but Romeo has, and he surely feels hopeless for he has lost the will to live.