Atomic emission spectra: classical vs. quantum mechanics
First of all, classical mechanics does not explain the emission spectrum at all, since there is no energy quantization out there. Bohr introduced energy quantization in his model in an ad hoc way by requiring the angular momentum to be an integer multiple of a fixed unit $\hbar = h/2\pi$ (which in turn resulted in energy also being quantized). So, Bohr model is not a classical mechanical model, since there is quantization of angular momentum and energy. But it is also not a quantum mechanical one since quantization of angular momentum is not justified, but rather asserted. 

I'm saying all this just to correct OP: it was not a fight of classical vs. quantum mechanics; classical mechanics was already knockouted and quantum mechanics didn't yet fully emerged. It was a battle between semiclassical-semiquantum theories and reality which today could be viewed as well as a fight between semiclassical-semiquantum theories and the real quantum theory invented later.

Anyway, there are in general many possible electron transitions for each atom (or molecule) that contribute to its emission spectrum and each transition has a specific energy difference and probability. The Bohr semiclassical model had shortcomings with respect to both these characteristics of electron transitions:


Multi-electron atoms do not have energy levels as predicted by the Bohr model, and consequently, the model predicts wrong positions of the spectral lines. In fact, it fails already for helium atom.
There is no general way of calculating probabilities of transitions, thus, it is not possible to provide relative intensities of spectral lines.