Dear Pies:
My answer is as simply as it can be: I love non-tonal (not just atonal, but pantonal, bi/politonal, modal and polimodal, serial, etc, etc, etc) music because it can express moods and feelings and things that tonal music can't. Furthermore, it is the label of an era, that is already opened: ours.
One example: Penderecki's Threnody. By no means one could depict the misery and disgrace of the destroyed japanese city in tonal music. Even a very chromatic minor tonality can not represent the horror of an atomic bomb falling, the massive death. When I listen to this composition, I always think about the strange and harsh world we live in, something that don't occur to me when I listen to Beethoven, for instance.
Of course, if you think about Webern's Bagatelles, or Berio's Sequenze, or any other great non-tonal work, you can find a wide palette of emotions and situations, that are neither better nor worst than the ones tonality can sing: they are different. That is, in my most humble opinion, the magic: we can listen to everything, from plainchant to McMillan, and every style is significative of its surroundings, which help us in the process of understanding so distinct times and places that are our heritage.
Best wishes!