The advantage of the odds formulation is that you can compute posterior odds in your head, whereas computing the posterior probabilities via the standard form of the theorem (that you wrote down here) involves an unpleasant normalization factor.

E.g. for the example in https://en.wikipedia.org/wiki/Base_rate_fallacy#Low-incidenc...:

    prior odds of infection =   2 :  98
  × likelihood ratio        = 100 :   5
  = posterior odds          ≈ 200 : 500

and thus the probabilities (if you actually need them at this point) are ≈ (2/7, 5/7).

See also 3Blue1Brown’s exposition: https://youtu.be/watch?v=lG4VkPoG3ko.