"Mass in Latin" vs. "Latin Mass"
It's important to understand that the term Latin Mass is almost always a colloquial and somewhat imprecise reference to the Extraordinary Form of the Roman Rite.
This is the form of the liturgy that, in its essentials, was used by almost all Western Christians from at least the early Middle Ages until the Reformation, was codified for the great majority of Roman-Rite Catholics in 1570, and has remained fairly fixed up to and including the latest edition, which was published in 1962.
The older form of the mass is relatively rare and hard to find these days (though it is enjoying a gradual resurgence) -- most Catholics instead use the Ordinary Form of the Roman Rite, which was first published in 1970 as the initial answer to the Second Vatican Council's call for a reform and revision of the Church's worship.
The official texts of both forms are promulgated in Latin by the Vatican, but the older form must, by law, be celebrated in Latin (with a tiny bit of Greek), whereas the newer form may be celebrated in Latin, but rarely is.
Hence the informal term "Latin Mass", which really signifies the ritual more than the language, but does so by way of the most immediately-obvious distinguishing feature.
Why anyone cares
The two forms are strikingly different. They share the same overall structure and have many elements in common, but also many significant areas of divergence in structural detail, ceremonial rules, vesture, and the particular texts and music prescribed for various parts of the mass and various days of the year. Typically, though not in every single case, these divergences culminate in a strikingly different atmosphere of worship at church on Sunday morning.
This is not perhaps the place to discuss whether one form is better than the other, but it's plain that some people energetically prefer one form to the other.