What is the importance of the mask in the Noh theatre?

Great question!
Noh Theater is a form of Japanese drama that originated in ceremonial performances at shrines and temples in the thirteenth century. Deriving its name from a word meaning “skill” or “talent,” Noh performances include music, dancing, and masked acting to communicate a Shinto theme by retelling well-known Japanese stories. Noh is a nuanced art form, and stage actors undergo intense training and preparation.
Masks are a vital part of Noh drama but serve a different function than many students of Western literature might presuppose. For many students of Western drama, theatrical masks evoke the tragedies of ancient Athens. In Athenian theater, actors wore masks to portray multiple characters and to exaggerate the emotions of a character for an audience that was often seated far from the stage.In contrast to Greek drama, Noh Theater actors do not switch characters during a performance, nor do Noh masks attempt to portray exaggerated emotions for the audience. Rather, the wooden masks, or omote, are carved with neutral expressions; performers attempt to imbue emotion to their character with slight head movements onstage. In this dramatic form, emotions are not communicated in the expressions of the face but rather through “minute movements” and body language. There are over sixty recorded varieties of masks used by Noh actors to depict elderly men, men, women, demons, spirits, and gods.
The relationship between actor and mask is highly important to the essence of Noh Theater:

In putting the mask on, the word kaburu (putting on clothing) is not used. Instead, the word kakeru (to hang) or tsukeru (to attach) is used. In this way, it is implying that the performer is “becoming” the mask, and its emotions, in order to better express the characters feelings.

In Noh Theater, the relationship between actor and mask is spiritual in intensity; the wooden omote are valuable heirlooms passed down for generations by actor-families. I hope this helps!
http://www.the-noh.com/sub/jp/index.php?mode=db&action=e_index&class_id=1

Noh is a highly expressive theatrical tradition. Unlike Western theatre, which relies heavily on naturalism, Noh is a stylized type of performance that aims to convey emotion rather than character. Masks are used to help illustrate the emotion. But unlike the masks used in Ancient Greek theatre, Noh masks are not just for the benefit of the audience but the performers. In modern Noh, each Noh performer has a mask unique to them, but these change with every performance. Although each mask is different, they tend to hold neutral expressions and are seen as symbolic rather than as ordinary costume pieces or props. The masks are also illustrative of a Noh performer's status, and not every performer wears them. There are different kinds of masks for the characters within a Noh drama, such as mortals, demons, and spirits.