Differences (and non-differences) between gender neutral and neutrois
Neutrois is more limited to gender identity, however some people used it as a descriptor of their sex characteristics tied to medically transitioning, as you can see in terms of MtN (male to neutrois) and FtN (female to neutrois), as well as non-binary and gender-neutral.
According to @neutrois (in neutrois.com), neutrois has no definition, except there lists hyponyms as supposed definitions, including neutral gender, null gender, neither male or female, genderless and agender.
Gender neutrality or gender neutral is synonymous to epicene or epicenity, gender-inclusive and unisex, specially when talking about language and pronouns. It's often used akin to or interchangeably with androgynous, agender, genderless, genderqueer, gender-free, ungendered, degendering, gender-fluid and non-binary, specially in fashion.
Just as my teacher says "current(ly) but not correct(ly)", the public use is not always technically right. Many people may feel offended or even dysphoric for seeing their gender identity used as a gender expression term, specially nonbinary specific ones.
Unrelated to the grammatical gender (neuter or neutral gender), neutral-gender or gender neutral identity is also a gender neutrality.
Many androgynes will describe their gender as unrelated to androgyny, just as multiple neutrois individuals do not associate with gender neutrality. However androgyne is received as a lingender or gender androgyny, just as neutrois is taken as a ningender.
Many times neutrois is incompletely described as having no gender or a specifically monolexic word for agender. Other times people explain it's different as in having a gendered feeling. Gender neutral neutrois people (whose gender is neutral, comgender or at least not exactly genderless) might feel unrepresented. But many people can't distinguish their own experiences from being gendered or nongendered, finding neutrois a useful term for that. And gender nullity or nultrois not always correlate for agender neutrois folx.
The meaning @neopronouns gave in eir post is not exactly incorrect. This graph exemplifies how many people see neutral as sitting in between male and female genders, but that happens with almost all nonbinary identities in a linear spectrum. Various gender neutrals fall outside these spectrums, some see it as a monolithic abinarity.
Neuter means neither (ne=not, uter=either) anyways. But many neutrois and gender neutral folks see themselves as none, all, both or both and neither.
Contrasting with genderpositive and gendernegative, genderneutral could be an ambivalence of the two, like being demiagendered, when your gender is present and not present simultaneously, not essentially a genderflux state. But genderpositive neutrois and gendernegative neutrois exist, just as gendernull neutrois and genderneutral neutrois.
But there's also neutral demigenders and saturated genders, being neutrois the central of the neutral spectrum, the balancement of all genders, and genderneutrality an abinary gender. The conception of neolingual Portuguese negative and positive (let's imagine like it's "negativx" and "positivx" in English) and their gender negativity & positive gender and gender positivity & positive gender bring neutre as a gender ambiguity, indifference or equilibrium, represented through zero (0).
Neuter sex and neuter gender/"n" are also used as a third or other gender options beyond "m"/male/man/masculine and "f"/female/woman/feminine options in formularies and documents, so that there will be even binary intersex, altersex, gender nonconforming and trans folks using it.