Willy misses this distinction because he is living in a dream world. He views his life as something that it is not. He sees himself as much more successful than he actually is. He has visions of grandeur that cloud his judgment and ability to think clearly about what and who he really is. His actions, such as having a mistress and not taking the time to really know who his sons were emotionally, reveal his lack of understanding about what it means to be loved. He is more worried about perceptions than he is the truth!
In "Death of a Salesman",Willy worries about how the outside world sees him.He wants to do well by his family,but his real obsession is that outsiders will like him.For example,Willy lies to make himself seem better to others.He pretends he is a much more successful business man than he really is.He tells Biff how important impressions will be when Biff gets older and has to go into the real world.Willy does not worry that Biff is not doing all of his schoolwork because he feels Biff will always be better off because he is good looking, talented and can tell people what they like to hear,very different from Bernard who is awkward, shy but very intelligent.Willy makes his opinions clear in the flashbacks that he feels his son is better than Bernard because of these facts.
If Willy was more concerned with being loved,he would worry more about his wife.Willy would make sure she had new stockings instead of his mistress.Linda, who has always been there for him even though times were tough,is put to the back because he is more concerned about what his mistress will think of him.Willy clearly lets the desire to be well liked rule every other aspect with his life, clearly making his family suffer because of it.
A key moment when this is seen is in one of the last scenes when Biff is confronting Willy. Linda says that Biff has always loved Willy, but Willy mistakes it for like which he thinks is more important. However, really Willy always wanted to be loved. Because of how strong his belief in the American Dream is he mistakes love for like. Also his guilt prevents him from accepting any love from his family.