Spectral bands are regions of a given spectrum, having a specific range of wavelengths or frequencies. Most often, it refers to electromagnetic bands, regions of the electromagnetic spectrum. 
More generally, spectral bands may also be means in the spectra of other types of signals, e.g., noise spectrum.
A frequency band is an interval in the frequency domain, limited by a lower frequency and an upper frequency. For example, it may refer to a radio band, such as wireless communication standards set by the International Telecommunication Union.