A great deal of Willie's sense of who he is comes from what he does. Interestingly, the title of the play is not "Death of a Husband" or "Death of a Father," but the very title shows how Willie sees himself. Although Charley's offer of a job, both early and later in the play makes great sense to the audience, Willie, although he is willing to take Charley's money, is not able to accept the job. To accept the job from his friend/neighbor, would be to admit that he is a failure and that all the "wisdom" that he gave his sons led to their failure. This is more than Willie can do. It's difficult to understand how taking the money does not do the same thing, but to Willie there is a clear difference.
It's probably very difficult to admit that what you always wanted to be probably wasn't worth the dream, and that you didn't achieve what you wanted in the first place.