When Dylan approached The Hawks in 1965, they were only vaguely aware of him. They came from a very different world, that of Rhythm & Blues, Blues, Jazz, Rockabilly (having served as Ronnie Hawkins’ band for a couple of years), and Hillbilly/Country. The Folk music heard on the radio was "p*ssy" music to them, effete urban and suburban "white" music.
They had been playing the "white chitlin" circuit, in working class bars across the east coast, midwest, and south (as well as up in Toronto), to hard drinking audiences, often separated from the audience by a chicken wire screen (to keep the flying bottles from reaching the stage ;-).
It was only after working with Dylan that they became what we hear on Music From Big Pink and the brown album, especially after spending all of 1967 in the basement of Big Pink, making music and recording it (the Basement Tapes, of course). Their influences---like that of Dylan---are deep and wide. Organist Garth Hudson is a classically-trained musician---reads music, knows music theory, etc. Levon Helm grew up listening to local Bluesmen---he lived in the same Arkansas town as Sonny Boy Williamson, sitting in the corner of the local radio station when SB performed live on the air---and The Grand Old Opry, just as had Elvis, Jerry Lee Lewis, Johnny Cash, Roy Orbison, and Carl Perkins. Bassist Rick Danko led his own Country band, once opening for Ronnie Hawkins & The Hawks. Richard Manuel played around Toronto, known for his Ray Charles-influenced vocal style and ability.
The Band were definitely not a Rock band, but they did play Rock ’n’ Roll (if you appreciate the difference). But much more: Hillbilly, Blues, Pop, and music which defies categorization via such crude terms.