Cates can be said to have "won" the case because the punishment ordered by the judge is a trivial one in comparison with the supposed seriousness of the charges against him.
In actual history, the Scopes "Monkey Trial" became a huge media circus, just like the fictionalized account in Inherit the Wind. In the play, though the authors take some liberties, the polarized climate of the time is accurately portrayed. Bertram Cates, in his teaching of evolution to schoolchildren, is seen by the conservatives as evil and as one who subverts religion and moral values. His attorney, Henry Drummond (the real-life Clarence Darrow), and the journalist E.K. Hornbeck (H.L. Mencken) are similarly viewed as atheistic and amoral threats to the "American Way." Prosecutor Matthew Harrison Brady, in real-life William Jennings Bryan, is a man who in his political career has invoked religion as the basis for a middle-American populism. Bryan and others opposed the supposed elitism of liberals promoting science as the basis of a new set of values. This was the "culture war" of that time.
When the judge hands down a relatively small fine as the sentence, the whole trial appears a trivial thing blown out of proportion to its actual importance. The outcome is thus a kind of moral victory for Cates and for progressives of his time.