Illusion is on the right track. I started with the NECO kit, too. The late Roger Johnston actually patented the process, but Wheeler engineering came out with a cheap kit and when NECO pointed out the patent violation, Wheeler pointed out that he was an attorney and could afford to keep NECO tied up in court for years, so NECO gave up. The problem is that Wheeler did some things differently from NECO and started a nationwide trend of making cheap kits and different methods of doing it, including junk moly powder, sprays and all manner of other things the quality of which is unclear for the intended purpose.
NECO uses laboratory grade, high purity moly that is extra fine. Enough of it to coat 100 bullets is about hlaf the volume of a BB, and even the small container they sell it in lasts for years and years. But it's not cheap. Cheap moly can have several drawbacks. One is that it is not so fine so it can build up small lumps in the barrel more easily. Another is that it can have more free sulfur that tends to attract moisture to create the acidic ions that initiate corrosion that leads to pits. It can also have free iron in it which helps promote corrosion. It can also have miscellaneous impurities that can make it abrasive, especially at high pressures. So a lot of people got started with moly using materials that tended to cause the problems described. I never had any of them using the NECO system.
The NECO process also coats the finished bullets with carnauba bean wax. This tends to keep it from rubbing off on your fingers. Other systems mostly don't bother. If you read the literature on moly, you find that if it is exposed to moisture it loses some of its lubricity, so I think the wax is good from the standpoint of maximizing the benefits of it.
The fact moly significantly reduces how fast copper fouling builds up in a bore has been misinterpreted by many to mean a gun using moly bullets didn't need cleaning all season, and would hang out at Camp Perry in 70%+ R.H. and complain moly causes corrosion later. High temperatures in the presence of oxygen and water vapor oxidize molybdenum and release free sulfur. Smokeless powder burns in an oxygen-starved reaction, so there the moly has to compete with the powder fuel atoms for that oxygen, and the reaction thus isn't bad while a bullet is in the barrel. Once it is out, if the barrel is warm and the gun is in humid air, you can expect some of this reaction to occur. Guns shooting moly bullets therefore need cleaning same as any other, or they need to be stored in very low humidity where rust does not occur (below about 40% for general purposes). But I haven't found they particularly need to clean a moly bullet gun more than any other, either, when the coating material quality is good.
The BoreTech product is probably the fastest cleaner. Gunzilla also works, but let it sit in the bore overnight before the final patch, as that gets it all out.
Finally, it takes more effort to adjust to using moly. Moly affects barrel time, increasing it if you don't adjust your load for the friction difference. This can move you off a sweet spot and make you retune a load. The bullets enter the rifling more easily, and slip down the tube more easily. This is why less copper fouling rubs off. But it also means they offer less reaction force for powder to build pressure against, have lower start pressure, and often lose 50 fps or so with a given load.
Moly doesn't just disappear from a bore without cleaning, so if you switch from moly to plain bullets without cleaning, it goes on to affect non-moly bullets, and if you switch between bullet types without thorough bore cleaning, you get intermediate results that don't reflect the behavior of either of those bullets in a barrel that shoots them exclusively. For this reason, the barrel needs settling time after a switch.
If you insert a moly bullet into a freshly trimmed and chamferred case, having done nothing else to it, then use a kinetic bullet puller to remove the bullet, you find the bullet bare of moly where the case mouth scraped it off on the way into the case. You can completely avoid that by burnishing a chamferred case mouth before seating a moly bullet into it. Bart B.'s recommendation to use the reverse thread of an E-Z out on a drill press to burnish case mouths is a good one.
There is evidence moly bullets self-center better in a rifle throat and the recovered bullets don't exhibit the little tails of gilding metal at the back edges of the grooves engraved by the rifling lands. This causes a very small, but measurable improvement in ballistic coefficient in most instances. There is less throat wear because of the pressure and temperature drop associated with direct substitution of moly bullets into a load, but since it is temperature and not friction that does most of the damage to a throat over time, the ability of moly to extend barrel life has been questioned and found hard to detect by some experimenters, but not by others. This result difference is likely due to variables that weren't controlled. With less friction, there should be at least a little less heat shooting moly bullets.
As to accuracy, it depends. If you have you moly load retuned for those bullets, there should be at least no loss of accuracy. If your guns is sensitive to bullet tilt, the better self-centering without distortion should improve accuracy. If your gun's accuracy deteriorates significantly due to heavy copper fouling, that can be prevented with moly. That's how I got started on it. My old DCM Garand's military barrel shot very well, right up to about round number 40, then would open groups up a point or two for the last rounds of a match. Switching to moly stopped that behavior completely. That gun had a very rough bore that accumulated copper quickly with plain bullets.