When considering the band structure of two materials or two structures along a specific k path, what exactly is meant by the gradient and the distance between band lines? Does this show any information other than the difference in the band gap of the materials? Is there a good reference book from where I can learn this?
The band structure determines the dispersion relation for the electrons within the material, the E(k) relation. Its gradient (the inverse of it, to be more precise) is related to the effective mass of the carriers. The occupation of the bands, according to the Fermi level, determines if the material is a metal or an insulator, depending on if the Fermi level crosses or not a band. More information about the electronic and optical properties can be obtained using the band structure together with the density of states. Also, the band structure can be resolved in spin, to elucidate magnetic properties. Therefore, apart from determining the bandgap, lots of information can be retrieved about the material's properties using the band structure.
Good references are Solid State books such as Ashcroft and Mermin - Solid State Physics; Gerald Bastard - Wave Mechanics Applied to Semiconductor Heterostructures; Charles Kittel - Introduction to Solid State Physics.
Hope it helps!