The significance of Hamilton's desire to defend an aesthetic conception of music lies in its past and present views. Music has historically been regarded as aesthetic, but only secondary to natural or mathematical or ethical. These distinctions take away from the actual intention of music as a means to an aesthetic end, and Hamilton means to bring that intention to the forefront, and define music correctly. Also, perhaps more importantly, music for aesthetic purposes has been viewed as natural, or godly, and thus outside the realm of human experience, whereas the human experience of creating music has been viewed not as an art, thus lowering its status and claiming it is not aesthetic in nature. Hamilton means to prove that practices of craft are still aesthetic in nature, that art and aesthetics are not always synonymous.
Yet Hamilton talks of music as being art with at least a lowercase 'a', which means to signify music's aesthetic intention. It is important to defend this claim because music is indeed a human activity with aesthetics at its core. As Hamilton says in his introduction, "...music is abstract in form, but humane in utterance - and utterance is essential." However previously or contemporarily viewed, music is and always has been humane in utterance and experience, and that speaks directly to determining the aesthetics of music.
And so, Hamilton's desire to articulate and defend an aesthetic conception of music is significant because this conception is actually the link between all the differing views of music, yet not always apparently, and it needs to be made apparent.
Is music merely craft? What is the distinction between a craft or skill, and art?