The reason that you hear the term so much now is that neoconservatives have basically carried the day when it comes to foreign policy (the more wars and foreign military interventions the better), and when it comes to big government (the bigger the better -- as long as neoconservatives are peopling the bureaucracies.)
They were or are intelectuals previously involved in Democrat policymaking when Democrats still though communism a bad thing. When George McGovern's comrades took control of the Democrat Party in 1972 and moved it in unacceptably radical directions, Lyndon Johnson gave private advice to Richard Nixon which helped crush McGovern at the polls. LBJ died of a heart attack at his ranch on the afternoon of 1/23/73 (Roe vs. Wade decision day). He scarcely lived to enjoy the results of his advice to Nixon. He did, undoubtedly, enjoy the election of 1972 immensely.
No one ever heard of what are now called "paleo-conservatives" until they suddenly discovered their own existence in about 1986 when it finally dawned on the paleowhatevers that Ronaldus Maximus wanted nothing whatsoever to do with most of them because they were policy eccentrics and social embarassments. They exploded in rage and rediscovered old pre-Pearl Harbor obsessive isolationism and an embarassing "blood and soil" dogma that sounds an awful lot like some other discredited crusades of the 1930s (central European style). The paleos claim an allegiance to the Constitution and to small government. There is no danger that they will ever be able to implement those ideas, assuming that they actually believe in them.
The conservative movement was NEVER "paleo." On foreign policy, the conservative movement was aggressively interventionist ever since Pearl Harbor made America grow up. Arts and crafts do not play a very large role in the conservative movement which leaves such matters to the taste of individuals so long as the arts and crafts are not abominations like Serrano's "P___ Christ" or the elephant-spattered Mary at the Brooklyn "Art" Museum. Conservatives are nationalists in foreign policy and unapologetically so. Conservatives fondly wish for free market success for ethnic minrities because nothing would be better for those minoriies' income or better for our nation's politics. Conservatives are quite conservative on social issues: babies, marriage and guns and place little confidence in gummint skewels. Conservatives favor cutting off the gummint's allowance by lowering taxes. While there are some conservatives who may dissent on an issue or two, the so-called paleoconservatives, like the leftist activists whose foreign policy and military views the paleos so emulate, have forfeited any legitimate claim to conservatism because they are reflexively cowardly in matters of military and foreign policy.
Because the conservative movement was reliably anti-communist, interventionist and pro-military, the actual neo-conservatives crossed the aisle and became Republicans to ally with us against the mush-spined moderates in the GOP. Most mush-spined moderates have long since defected to the Demonrats because they are joined at the hip with Demonrats on social issues (and good riddance to them), a handful of "paleo" publications, institutes and websites survive to disrupt the GOP and give aid and comfort to the enemies of the GOP and the enemies of the USA.
You will find most uses of the term "neo-conservative" as a substitute for "conservative" in left-wing publications, speeches and calls from C-SPAN commies in the morning, from so-called "paleo" sources and from people who wax nostalgic for the "good old days" when Democrats outnumbered Republicans by massive margins in FDR's heyday, from the enemies of the conservative movement (enemies of Reaganism) and of our country and from the usual gang of racist and anti-Semite clowns. One who uses the "neo-conservative" slur may emerge from any of such backgrounds without being in sympathy with any or all of the others. The users of the term "neo-con" are essentially a coalition of losers anxious for the USA to lose.
Of course, even the paleos are right once in a while. Conservatives ARE a LOT smarter than paleowhatevers. Being a lot smarter is why the conservatives have carried the day politically and particularly on foreign policy and military matters.