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Ultimately, they are just different expressions for the same fundamental physical quantity: the energies of quantum states and the differences between them. There are different units you can use to express that quantity - energy, frequency, wavelength, wavenumber - but at the end of the day they are all equivalent, and can be converted among each other, as $$... 3 The method they are using seems to be looking at the separation between two different transitions to determine B. J'' denotes the lower energy state in a transition and J' denotes the higher energy state. Here, you are looking at two different transitions:$$J_1'=4\to J_1''=3 \text{ with } \nu_{4\to 3}=\pu{15.365cm^-1}J_2'=9\to J_2''=8 \text{ ...