First, the prediction of Neptune was a big win for science in the eyes of the general public. It was not exactly spotless though, especially in the eyes of the scientists. Here is from Kelley's How was Neptune Discovered?:
"The world was excited by the find, for never before had mathematics predicted a natural object. This confidence in the results was lessened, however, when discrepancies were noticed in the calculated values and the actual ones (Lyttleton 227). For example, Adams calculated an orbital period of 227 years and Le Verrier found it to be 218 years using Kepler’s Third Law (Period squared is proportional to average distance cubed). The actual value of the orbit is 165 years. This discrepancy was not a result of using Kepler’s Third Law but because of using Bode’s Law for the average distance (229). The only actual value they were close to, if one looks at the table, is the location in the sky it would be found. It is possible that both men were simply lucky with this. We shall never truly know (233)."
Second, debate over evolution did not begin 14 years later, it was going on for a while Lamarck proposed his version of evolution in 1801, and even the dominant in the early 19-th century Cuvier's catastrophism posited a kind of evolution, although in discrete leaps. Lyell's gradualistic geology defended in Principles of Geology that displaced it was far more influential in the acceptance of Darwin's ideas than astronomical events (and it was a direct inspiration for Darwin himself). In 1860 Leverrier himself moved on to a new prediction, Vulcan, blamed for the anomalous precession of the perihelion of Mercury. A curious report about the annual meeting of the British Association for the Advancement of Science in Manchester, 1861 indicates that Neptune was not much of a factor:
"In 1861, the city showcased both its wealth and brains as it hosted Britain’s largest celebration of knowledge, the annual meeting of the British Association for the Advancement of Science. Charles Darwin had published The Origin of Species less than two years before, and that explosion continued to reverberate through every gathering of the learned. At the Manchester meeting Darwin’s defenders prepared to battle religious doubters. One speaker, the “blind economist” Henry Fawcett, made the ultimate claim: Darwin was a true scientific hero, one who solved his problem by the same methods, the same approach to experiment, observation, and generalization that the great Isaac Newton himself had used in his physics. Much else was discussed, of course — advances in dredging engineering, a report on birds of New Zealand, news from the balloon committee. The astronomy section was relatively quiet..."
Finally, in contrast to 17th and 18th centuries the role of the Church in scientific debates was much diminished, and its reaction to Darwin (and more specifically to human origins, not evolution in general), was ambivalent:
"The concentration of concern on the implications of evolutionary theory for the human species was to remain typical of Catholic reactions. No Vatican response was made to this, which some have taken to imply agreement. No mention of evolution was made in the pronouncements of the First Vatican Council in 1868. In the following decades, a consistently and aggressively anti-evolution position was taken by the influential Jesuit periodical La Civiltà Cattolica, which, though unofficial, was generally believed to have accurate information about the views and actions of the Vatican authorities. The opening in 1998 of the Archive of the Congregation for the Doctrine of the Faith (in the 19th century called the Holy Office and the Congregation of the Index) has revealed that on many crucial points this belief was mistaken..."
