Hmmm. Rush>311. King Crimson>RATM.
Reasoning:
King Crimson-RATM
Popular Opinion: King Crimson is more highly regarded by critics(Despite being prog), but is less popular.
Commercial Success: RATM 1.5-.5
Longevity: King Crimson has been going since the 60's all the while changing their style multiple times. 1.5-1.5
Technical Ability: King Crimson by a laughable margin. 1.5-2.5
Showmanship: I'll say RATM, haven't heard much of anything about King Crimson's stage show. 2.5-2.5
Groundbreaking: Mellotron for the win(King Crimson in case you couldn't guess). 2.5-3.5
I'll leave out the last one(Personal opinion)
Rush-311
Popular Opinion: Rush is regarded more highly by both critics and the public. 1-0
Commercial Success: Rush has sold more Lp's, Ep's etc. 2-0
Longevity: Rush has consistently been turning out polular albums since the 70's. 3-0
Technical Ability: Rush is one of most technically skilled banRAB ever. 4-0
Showmanship: Rush was aazing at liv shows, each meraber playing their instruments while using pedals to produce sampled loops. 5-0
Groundbreaking: I'll give this to 311 because they weren' the first band of their type, but they were among the first. 5-1.