First, let's start what they have in common:
1) They both evolved, soundwise, throughout their careers.
2) They both followed their own arrows in terms of what they wanted to sound like.
3) They both had "greatest of all time" musicians, but not for the same reasons.
4) They both used any technology that made sense to make the music they wanted to make.
5) They both made some concept albums.
6) They had profound interconnection/interplay between musicians in the band. (Gilmour/Wright, Alex/Geddy/Neil)
7) You could probably but some labels on their genres that overlapped.
8) They both had multiple Magnum Opus albums.
What they had that was different:
1) The scope of the visual, technological, and aesthetic of Floyd's performance far surpassed Rush's.
2) Floyd was, in general, more serious, and more storytelling-through-music oriented in a way that was less intense.
3) Floyd's sound is likeable across genres to a broader variety of people.
4) Rush was a considerably more hard touring band, reaching more people live.
5) Floyd had a GOAT level guitarist in Gilmour. People across genres aspire to sound and play like him. People who love Rush love Alex, but he isn't a Rushmore guitarist.
6) David Gilmour has a vocalist sound that is not polarizing in any way. He may have one of the most soothing, palatable lead singing voices in rock history.
7) The members of Floyd were different enough to play each other's foils, musically, character-wise, and on stage. This is part of the reason they had so much drama.
8) Rush was considerably more technically oriented. There was probably no musician in Floyd that had the virtuosity of Peart or Lee.
9) Rush were sillier. This is a difference, not a pro or con.
10) Rush was, on average, considerably more energetic in their sound and delivery.
11) Floyd's Magnum Opus albums were bigger in scale.
So, those are the differences in my mind. Most great guitarists, even ones consider icons, wouldn't be considered as universally great as Geddy Lee... except, well, guys like David Gilmour. And for Gilmour it was never about technicality, it was about style.
Floyd's albums just penetrate the soul in a different way.
I love Rush though.