Band structure is referred to a bulk system with periodicity. For the more details, you had better read a textbook on the solid state physics to understand the basic background.
Molecular energy spectrum---> the digram of energy levels of a molecule or an isolated system (e.g., cluster).
The band gap can be read from the band structure by looking at the bottom of conduction band and the top of the valence band. If you want to read the band gap from the transmission spectrum, the transmission coefficient at E_F should be zero, and then you may look at the first energy above E_F with non-zero transmission coefficient and the first energy below E_F with non-zero transmission coefficient.