Why did Italians become "white" but Latinos did not?

I take some issue with the premise of this question. 'Latinos', meaning, as I understand it, Spanish and Portuguese speaking people who inhabit the Americas from the US to Antarctica, are a very racially and ethnically diverse population. Included among them are many people who are considered 'white' and who have historically been categorised as such.

In the Spanish colonial system, whiteness was never formally defined, but there was an implicit understanding that it existed because every type of racial mixture aside from white was defined, and carried with it certain legal barriers or outright disadvantages. While there was a lot more intermixture between native, African, and European populations in the Spanish and Portuguese Americas than in the British Americas, notions of whiteness absolutely still existed and still exist today, and by the late 18th century the term was being used often informally to refer to Europeans who had less than 1/8th admixture with non-Europeans.

So, if I were to answer your question in a sense that does not take the US as the centre of the world, the answer is 'many of them always were and still are considered white.' From the very development of the concept of whiteness in the Spanish and Portuguese Americas, it was used exclusively to refer to people who would be considered 'Latino' today, and whose descendants still are considered 'Latino' today. The landholding elite in Latin America, which in many respects still dominates their respective countries' economics and politics, has always been mostly white, both in terms of self perception, legal categorisation (at least through implication), and through perception by others within their societies.

It is a uniquely American thing to group all of the inhabitants of Latin America into a single racially imprecise group, regardless of how they look and how their families benefited from racial classifications in the past, seemingly based on nothing aside from their mother tongue and where their ancestors lived. This conception has its roots in American conflicts with Mexico, which brought with it a lot of dehumanisation of people from former Spanish colonies, regardless of their race. The subsequent annexation of Mexican territory then led to legal and extralegal discrimination against those who stayed in the new US territory, despite the fact that many were granted citizenship. White Mexicans who had for centuries been considered 'white' in the Spanish Americas suddenly found themselves in an environment where it was often convenient for their new rulers to deny them that privilege. That said, legally speaking, Mexicans in the US - whether they had been considered white or not under the Spanish colonial system - were mostly grouped into the 'white' category unless they were active members of an Indigenous group or were black.

I'll leave it up to someone else to go more in depth on the unique conception of whiteness in the US that led to today's weird situation re: 'Latinos'. But the short answer is that at least some 'Latinos' were always white, and that status afforded them similar privileges to those enjoyed by 'whites' in the British North American colonies.

The first part of your question also merits some clarification. There is a pervasive idea that Italians and other European groups were discriminated against due to the idea that they weren't white, but this is mostly an exaggeration. From the very beginning when whiteness became legally defined in the British American colonies in the 17th century, all Europeans were tacitly included, even the Irish and certainly the Italians who came much later. In fact, one important factor leading to the legal codification of 'whiteness' in colonial Barbados and Jamaica, which was then adopted in North America, was that Irish indentured servants were participating in violent resistance alongside African slaves. By legally including the Irish in the new dominant 'white' group and progressively giving indentured servants more and more concessions, divisions were created that disincentivised further collaboration between white indentured servants and African slaves. The Irish continued to endure pervasive discrimination, but it was based on their Irishness, their Catholicism, and their status as victims of English colonialism, rather than them not being white. Legally, they were considered white - it was only in some social circumstances where they were excluded from the category. This applies equally to other groups of European migrants who endured similar discrimination later on.

Discrimination against non-Anglo-Saxon/Protestant European migrants was rooted much more in hierarchies among white people, rather than the conception that they weren't white. The USA had its roots as an English Protestant colonial project, after all.