Uranus was the first planet to be discovered telescopically, while Neptune was the first to be discovered mathematically. But, oddly, both had been previously observed without being recognized as new planets.
William Herschel discovered Uranus by chance in 1781 - until then people didn't think there were any planets beyond Saturn. After several decades of observation astronomers realized that Uranus wasn't moving as expected. French astronomer Urbain LeVerrier thought this was due to the influence of an undiscovered planet. His calculation of its position enabled Johann Galle at the Berlin Observatory to find it in 1846.
Herschel wanted to name his planet after his patron George III, but German astronomer Johann Bode suggested Uranus, Roman sky god and father of Saturn. This maintained the classical theme, as did Neptune, named after the Roman sea god.
Uranus and Neptune, like the other two gas giants Jupiter and Saturn, have no solid surface. Descending through the atmosphere, the pressure increases until the gases are liquefied. But unlike Jupiter and Saturn, between the atmosphere and an Earth-sized rocky core, there is a mantle consisting of water ice and other ices. Therefore Uranus and Neptune are also known as ice giants.
The atmospheres of Uranus and Neptune are predominantly hydrogen and helium, but with 1-2% methane and also traces of water, ammonia and other compounds. Methane crystals give the planets their blue hue because it absorbs the red light and reflects the blue.
These distant twins are similar in size - about 4.0 and 3.9 times the diameter of the Earth, respectively - and their days are of similar length, about 17.25 hours for Uranus and just over sixteen hours for Neptune.
In addition, each has a ring system and multiple moons. Uranus has fourteen known rings and Neptune six, all made of very dark material. Uranus has 27 known moons and they are, unusually, named for characters from Shakespeare and Alexander Pope. Neptune has 14 moons, the largest of which, Triton, is thought to be a captured Kuiper Belt Object.
However they aren't identical twins.
Neptune's axis is tilted about 28 degrees. The axis is an imaginary line connecting the poles, around which a body rotates. Earth's axis is tilted about 23 degrees, so Neptune would have seasonal variation similar to Earth's, though the climate is obviously less agreeable.
Uranus however is tilted at 98 degrees, which means that it's orbiting on its side. If it's midsummer in one hemisphere, the other is entirely dark. Even though overall the poles of Uranus receive more solar radiation than the equator does, its equator is still warmer. We don't know why and its great distance makes it difficult to study.
Most of what we know about the twin planets is from the Voyager 2 mission in the 1980s. However during spring equinox in 2007, both Uranian hemispheres were in sunlight, presenting a good opportunity for observation.
Yet another surprise is that both planets have temperatures of -220 degrees Celsius (-365 degrees Fahrenheit) at the cloud tops. Uranus is nineteen times farther from the Sun than Earth is, but Neptune is thirty times the distance, so it should be colder. In common with Jupiter and Saturn, Neptune has some kind of internal heat source, because it radiates more heat energy than it receives from the Sun.
Although both Neptune and Uranus have cloud bands parallel to their equators, these are much more pronounced on Neptune. Neptune's atmosphere is very active, showing rapidly changing patterns, including the highest winds of any Solar System planet--up to 2000 km/hr (1200 mph).
(1) J J O'Connor and E F Robertson "Mathematical Discovery of Planets"
(2) "Neptune's Rings and Moons" http://www.windows2universe.org/neptune/moons_and_rings.html (accessed 11.01.10)