Unless you are a musician, there is always going to be a technical aspect of music that you will be unable to appreciate.
And that's fine, it applies to virtually any field of endeavour. Visual arts, architecture, medicine, sports, you name it.
But it does explain how some bands end up popular among musicians and yet don't have traction among non-musicians.
While it's certainly beating a dead horse, a lot of the support for Rush comes from the musician community. There's a lot about Rush that is easy to dislike, often Geddy's voice, some of the cheesier lyrics, but in terms of technical proficiency, there's a lot to appreciate and learn from.
As someone who has been much more actively playing music in a collaborative fashion over the last 5 years or so, and getting into steady gigging and performances with fellow musicians, how I listen to music has changed fundamentally from pure sonic appreciation into the mechanics of how I would perform the song with my band, what parts of the song would be more complex to interpret, how I would chart it for different instruments and voices.
There are songs that I thought were easy to pull off that were actually deceptively difficult and vice versa. After trying to write lyrics for some original material, I have a new appreciation for how difficult it can be to turn a phrase in a song, and I spend a lot more time listening to the words than before.
It's not better or more important IMO, it just adds another layer in terms of interacting with the artform. I can't turn it off, so it's just part of my filter at this stage.