What is the difference between Neutral Evil and Chaotic Evil?

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I suppose a problem I have is that it seems Chaotic Evil is often seen as an "insane" alignment — i.e., if you're Chaotic Evil then you have to be some sort of mindless, frothing-at-the-mouth lunatic who attacks everything around them at all times just because… I mean, after all, that's the alignment of demons, right? …forgetting that it's also the alignment of red dragons, and I figured they're aligned as such due to seeing themselves as being all-important and the center of everything — "me me me," and all that.

So on a whole to me your character seems much more NE than CE but I suppose that is a matter of opinion.

One of the major issues in d&d is comparing a CE mortal to a demon isn't a fair comparison. Demons are embodiment of chaos as well as evil with a nice dose of what appears to be madness to the mortal eye mixed in. The real point is that since they are made of evil chaos they fundamentally don't process information the same as a mortal. So everyone uses them as a yard stick and says CE = madman who can't make a rational but enjoys killing. When sure that madman might be CE but he isn't CE because he is crazy. It is the same reason that the Chaos alignment doesn't automatically make anything it is attached to crazy. Disassociating mental disorder and alignment is pretty important to understand alignment.

I mean, all of these seem pretty illegal, though, so they'd all be chaotic. They do seem to go in increasing order of evilness, but Chaotic Evil is not more evil than Neutral Evil.

Also, who would buy a random baby from a complete stranger?

I mean it is only illegal so much as doing so is considered illegal in said society. I mean take a modest proposal (anyone who hasn't read it should it is quite interesting) if a society took up the practice of eating babies because it is an easily available food source and other food sources are very scarce; a baby would then become a standard commodity in said society. As such the trading baby for apples and selling a baby to a strange would be perfectly lawful acts, though I can't see the kidnapping and holding it for ransom to ever being lawful.

The opposite of Chaos isn't so much Law, as Order. It is possible to indulge massively in illegal acts, in an orderly fashion.

A crimelord who runs an extremely organised criminal organisation, can be a pretty good exemplar of LE.

Even archdevils (about as Lawful as LE gets) can be crimelords. Glasya was one in 5e.

I disagree law seems just as important as order for the lawful chaos spectrum. But in general I agree with you, for example the leader of a terrorist group would most likely be CE; however, many of the zealot followers of said terrorist group would probably be LE. Strangely for me I have issues picturing Chaotic characters being a part of a group unless they are the leader of the group.