Neptune and the other gas giant planets emit more energy than they receive from the sun. They're not hot enough to undergo nuclear fusion as stars do, though. So that extra energy has to be coming from somewhere else.
Neptune is supposed to have a solid core surrounded by methane and other hydrocarbons in an ice-like layer, beneath an atmosphere consisting mainly of lighter elements hydrogen and helium. By mass, the planet is 10 to 15 percent methane, and the stuff exists at pressures greater than 20 gigapascals (much greater if you go deep enough).
Theorists had predicted for some time that methane (CH4, a molecule consisting of 1 carbon and 4 hydrogen atoms) could dissociate into hydrogen and carbon at high enough temperatures and pressures. The heavier carbon (of which diamond is composed) would fall to the planet's core, its gravitational potential energy being converted into kinetic energy and, in turn, to heat as a result of friction with the fluid it was falling through. If this happens at a high enough rate, you've got a substantial source of internally-generated energy for your gaseous planet, maybe even enough to account for that emitted by Neptune, about double what the sun sends it.
In 1999, a group of researchers at the University of California, Berkeley compressed methane to 10 to 50 gigapascals in a diamond anvil and heated it to 2000 to 3000 kelvin with a laser, with the intent of replicating conditions found in Neptune. At the point of direct interaction with the laser, diamond (pure carbon) formed, and a variety of exotic hydrocarbons showed up in the surrounding area.
The researchers published their findings (Science, vol. 286, pp. 100-102, 1 October 1999) with suitable caution, never claiming their results actually represented Neptune's atmosphere. Berkeley publicists weren't so subtle: the press release was titled "UCB Researchers Find Diamond Showers on Neptune and Uranus." Never mind that the misrepresented scientists didn't actually look at Neptune and Uranus with a space probe or even a telescope, and hence couldn't possibly have found anything there. Apparently, the more honest "UCB Researchers Precipitate Diamond in Laboratory Simulation of Neptune" just doesn't make good copy.
Neptune has many molecules in its fluid interior aside from methane. There's water, ammonia, and probably ethane and other heavier molecules as well. All these need to be taken into account when modeling the planet's chemistry. The Berkeley study was both fascinating and highly suggestive of diamond formation, but it isn't proof of what exactly is going on there. And until somebody gets conclusive data from Neptune itself, we can never really be sure.