Progressive Rock as a genre is very nebulous. A measure of virtuosity in skill and an eclectic sound are largely what characterizes one's perception of a band as being progressive. Because of that and many prog bands' tendencies towards genre fusion and a general interplay between genres mean many bands could be labeled as such. In a way it serves as more of a bridge genre than anything. A musical lingua franca!
Rush is one of the more popular prog bands because, compared to many others, they are much more accessible to people who otherwise wouldn't listen to prog rock. The complexity, themes, and overall sound can turn most first time listeners off entirely. Give the uninitiated King Crimson's "In The Court Of The Crimson King" and Rush's "Moving Pictures" and Rush wins every time. It's less intense on the ears.
Rush, Yes, and post Gabriel Genesis are "foot in the door" progressive. They retain many of the hallmarks of a prog rock band, but are easier to listen to and are more radio friendly. How often do you hear King Crimson, Procul Harum, or Gentle Giant on the radio? Not often I'd bet. I've heard King Crimson's Dinosaur maybe once. In fact, most prog mainstays adopted a streamlined and poppier sound in the 80s. Album long suites that trudged along became less common, which also helped to soften peoples' image of prog rock as being pretentious and overly grandiose. We could argue about whether or not doing either was "selling out" but you can't deny that it exposed them to a wider audience.
Someone who felt a band's music was incomprehensible before could sample their more familiar sounding offerings, dipping a toe in the shallower parts of the genre. When they finally jump into the deep end, they have a sense of what to expect in terms of composition and content. It becomes less intimidating and easier to grasp. When learning math you don't start with calculus, but with basic arithmetic and work your way up. Acquiring a taste for prog rock is the same way, but backwards. You start with pop, not its roots in psychedelic rock.
Regarding Rush and what makes them a progressive band, the first and most obvious is their ability and instrumentation. Without getting into the minutia of music theory, they have both great skill and a wide range of instruments in their body of work; especially Neil Peart. A second example is the synthesizer. The birth of progressive as a genre and the development of the synthesizer happened around the same time and progressive bands were some the first to incorporate them and other electronic sounds into their music. Rush was notable for putting it front and center in many of their arrangements in the 80s.
Another key trait was their exploring and adapting other genres into their music. Not just classical, but jazz, reggae, new wave, and even alternative rock to name a few. It's actually through Rush and prog rock in general that I developed an appreciation for jazz. I can't say I enjoy jazz but I least better understand and appreciate what jazz musicians were trying to do.
In terms of Rush's earlier influences you can count Yes, King Crimson, Genesis, Jethro Tull, and even Led Zeppelin as some of the bands they've said were inspirations to them. They started with a "traditional" progressive rock framework and grew from there. You could say they progressed.
