Uranus was one of the "original" planets known to all ancient civilizations. It is not known who discovered it first, as records for it exist in charts from many civilizations. Neptune, however, was the first planet discovered in the "modern era" -- and by discovered I really mean predicted. It has been known for some time after Newton, and his laws of planetary motion and universal gravitation, that orbits of celestial bodies are really eclipses, not perfect circles, with the sun being one focus of the elipse. It was also known to him, and those before him, that bodies move faster when closer to the sun; slower when further out (Kepler, I believe discovered this, but Newton was able to derive the same conclusion from calculus). Nevertheless, the orbit of Uranus was always off; not where it should be. It suggested that were was another planet beyond it that was tugging on its orbit, and thereby causing devations from it's expected observations. So, pouring over star and planetary charts lead to the prediction that if such a planet did exist, it could only be in one particular region of the sky. William Herschel would win that race and find Neptune, very near to where it was predicted to be. So, what really drove the discovery of Neptune was smart people using the best models of planetery motion at the time, noticed some oddeties and wanted to be the first to discover why their observations didn't match predictions.
