Irving Kristol, Bill Kristol's father, is often regarded as the godfather of Neoconservatism. He used the term in the title of various books and essays, applying it to himself and his end of the political spectrum.
Neoconservatism has roots in the Cold War Liberal wing of the Democratic Party. Hawkish on foreign policy and the use of force, and liberal to socialist in domestic policy. It's their domestic big government liberalism that often grates on the nerves of traditional conservatives.
Yes, the post that started this argument was the one in which I pointed out that the libs hate the neocons much more than traditional conservatives precisely because the neocons are most like them and therefor (according to the liberal mindset) more dangerous than pure Conservatism, which the libs don't believe is a credible danger. And they are probably right.