Everything that follows is all based on the original link in the original post. There are too many different listings and interpretations of the traditional alignments to make any argument based on other source materials. The OP provided a link to the material he was using to form his opinions, so no valid argument can be made to express the differences between the two alignments without referencing the original source, and building an argument and opinion based on that.
Everything else is an opinion based on a different interpretation, and illustrates nothing but someone else's interpretation of someone else's interpretation.
So, what follows is what I have gleaned from the original source as the differences between Lawful Neutral, and True Neutral, with examples extrapolated as probable manifestations of these principles in a character.
So, using the link you provided in the original post, True Neutral may do "what seems to be a good idea." This could be anything from helping someone pick up their groceries to killing an injured animal, or setting a house on fire while someone perceived as dangerous is trapped inside. If, at the time, it seems like a good idea, the character may do it.
The link also says this version of True Neutral "doesn't feel strongly one way or the other when it comes to good vs. evil or law vs. chaos. Most neutral characters exhibit a lack of conviction or bias rather than a commitment to neutrality." This means they would also lack enough conviction to support following any kind of inner law or set moral code. This would be one of the major defining differences between Lawful Neutral and True Neutral; one follows the "law" of a personal code or convinction, and the other lacks enough conviction to build a code to follow.
It goes on to illustrate a version of True Neutral that represents more of the philosophical essence of neutrality, itself. "They see good, evil, law, and chaos as prejudices and dangerous extremes. They advocate the middle way of neutrality as the best, most balanced road in the long run." This means a character identifying as this version of True Neutral will aid those marching against tyrany if tyrany represents the extreme. This character may also turn on the heroes they just helped and kill their leader if the leader represents an unfair advantage given to another extreme, like Law or Order. They may join rebels stealing, killing, and invading areas where lawfulness is perceived (by the character's own sense of balance and neutrality) as being an extreme. They may help children escape slaughter.
What they are unlikely to do is feel obligated to save children in a building burning for reasons other than someone evil setting it on fire intentionally. They are unlikely to feel obligated to help old ladies carry groceries home. They are unlikely to feel obligated to help a theif break into a house, or an abusive husband beat his wife, but only because they perceive no shift in balance or power in any of these acts.
If they perceive the theif is breaking into the house of someone with an artifact capable of amplifying their tyrany, they may help the thief. If the man beating his wife is beating a woman the character perceives as being too virtuous -- someone capable of tipping the scales too far in the favor of Good, or Law--, they may help the man beat his wife.
A True Neutral character -- whether taken in the form of someone who lacks enough conviction to define their own moral code, or as someone whose moral code is that of ensuring the world exists exactly in the center of the scale --, is one whose actions are defined entirely by personal perception from moment to moment.
The Lawful Neutral character, as defined by the original post's link, is about regulation. It is order and predictability through neutrality, rather than a biased moral code based on either Good or Evil, or balance, or impartiality.
Lawful Neutral is not for the betterment of mankind, or the downfall of mankind. It exists to rigidly uphold and defend order and regulation, regardless of who is making the laws (Good or Evil), and to unify all under the banner of harmony they believe is provided by that order and regulation.
According to the alignment detailed in the link, Lawful Neutral exists purely for the sake of Law. Their neutrality regards not whether something is considered Good or Evil, or whether a law or action falls outside of a balance between Good and Evil. Their neutrality addresses the lack of concern they have for whether a law they enforce is considered Good or Evil.
Taking this down to a personal level, rather than a governmental level, it means the character has a moral code or personal law which guides their actions, which they defend and enforce regardless of whether their actions or beliefs are perceived by others as being Good or Evil.
This is a character who may mutilate the bodies of their fallen foes, without fail, because they believe the mutilated body prevents the spirit from being accepted into the afterlife. Their personal law may assign all foes a state of ignobility by virtue of standing against their personal goals, and by that qualification alone, earn them denial to the afterlife. Any being or creature misguided enough to stand against their righteous will is, by necessity, unworthy of an afterlife, which it is their duty to see obstructed.
This is a character who may group with others, only to turn them in to relevant authorities when those authorities are met if those they grouped with have broken regional laws.
This is a character who will not kill the innocent, but also may not defend the innocent if the innocent are being attacked by regional authority figures, even if it is not clear what the innocents had done to receive the attention of the law.
Emotionally impartial, their loyalty is to order, regulation, law (in whatever form it takes), and the harmony that comes from everyone following the same rules at the same time. They do not question authority. They do not think independently to ask why a law is or isn't. They do not consider the consequences of the laws, or the fate of those who break them.
Law is law.
I think it would be incredibly interesting to actually write about a Lawful Neutral character ripped away from the regulation and order they know in such a way they are not allowed to return. Forced to group up with others, it first appears they must grow and adapt as people through embracing their new, more loosely structured lives, but instead they turn on those who aided them at the first opportunity to join law and order again, regardless of where or how it manifests. Because it fits the Lawful Neutral archetype, and I don't think it's been explored very well, or very often.
Anyway, there's your breakdown of Lawful Neutral vs. True Neutral, as detailed by your original link.
Maybe this finally answers your question.
