Face, brave, challenge, dare, defy, beard are comparable because all carry the meaning to confront with courage or boldness.
Face carries no more than this general sense; basically it suggests the confrontation of an enemy or adversary, but in its extended use it implies a recognition of the power of a force, a fact, or a situation which cannot be escaped to harm as well as to help and a willingness to accept the consequences.
Brave may imply a show of courage or bravado in facing or encountering. More often, however, it implies fortitude in facing and in enduring forces which ordinarily would strike the spirit with terror.
Challenge generally implies a confrontation of a person or thing opposed in such a way that one seems an accuser imputing weakness or fault in the one confronted. Often it may lose the feeling of accusation and then may mean no more than to dispute or question or, on the other hand, it may go farther and suggest a bold invitation to a contest (as a duel or other test of rightness or skill) which the one challenged cannot refuse or it may suggest bold measures inviting a response or retaliation.
Dare also usually emphasizes boldness rather than fortitude, but it rarely suggests the critical or censorious attitude so frequently evident in challenge. Rather, it implies venturesomeness, love of danger, or moral courage and may connote great or especial merit or mere rashness in the action.
Defy, like the others, usually implies a personal agent, but it may be said of things as well. When the idea of challenging is uppermost, the connotation of daring one to test a power which the challenger believes undefeatable or to do what the challenger believes impossible is usually its accompaniment. In either case there is a stronger implication of certainty in one’s belief than there is in challenge, and often a clearer suggestion of mockery.
When the idea of resistance is uppermost, there is a suggestion in defy of a power to withstand efforts, opposition, or rules. It is in this sense that a personal agent is most often not implied, for resistance does not always suggest an exercise of will.
Beard, although it implies defiance, often differs from defy in suggesting resolution rather than daring or mockery as its motive; in that way it comes somewhat closer to face and to brave.