Two points I'd like to make.

1. It's pretty impossible to answer questions about why something didn't happen. The answer is always basically "because it didn't?". With the exception of things that were actively presented which were otherwise on course to happen, we can't really address why something that could have happened didn't, because the possibility was only ever just that, a possibility.

2. It did? So now everything that follows deals with that but I'm leaving the format of numbered points.

The difference between a language and a dialect, speaking in strictly linguistic terms, is this: There isn't one. Not at all. Not even remotely. There are people who will argue for a way to define such a distinction, the most common line drawn being about mutual intelligibility. This is a nonsense line to draw, and here's why.

Mutual intelligibility is not a quantifiable feature of language varieties. You can not measure it consistently across situations because, first of all, how would you even quantify the high end of that without having an infinite number of hours of corpus data to work with, and more importantly, because people are incredibly subjective when reporting intelligibility. Motivations of the listener must be taken into account, and often what people report is not actually true were the issue to be pressed. Maybe I'm some high prestige high class snobby purist who just coincidentally happens to speak my country's standard dialect, and you're some slob of questionable pedigree speaking what I'd consider a low brow corruption of my great tongue. I will likely not admit to understanding you as well as I actually probably can, because it's a matter of class differentiation and identity preservation on my part to deny understanding that might actually be there.

So mutual intelligibility is out. How else do we draw the line then? Another common way is to say well ok this dialect has an army and a navy. I hate that quote. I loathe people who post it as the answer to questions like this on Reddit, because they are either entirely missing the point of the quote, which was tongue in cheek to begin with, or they think it's an actual answer. It is not an actual answer. Plenty of languages, like Basque, are clearly languages according to most people, yet do not have a nation behind them. So that's not an answer.

Then how do we draw the line? Well, as linguists, we don't. When I'm at a linguistic conference giving a talk on my dialectological study of under-documented varieties of Tibeto-Burman, you can be damn sure I still say the word "language" and "dialect", but I do so will full understanding that these are ultimately BS terms, and with the full understanding that the person I'm speaking to also has that understanding that they're ultimately BS terms. It's short hand, because short hand is useful, but it's short hand for something that we have a shared understanding about and we're just going for expediency. What I don't do is put those words in my formal publications, because there it matters. Instead, everything is a variety. English and German are varieties of Germanic. American English and Aussie English are varieties of English. Melbourne English and Queensland English are varieties of Aussie English. It's varieties all the way down.

All that to say, Latin didn't split into a bunch of different languages, or to say that Arabic did split into a bunch of different languages. The distinction is purely extra-linguistic, and fundamentally socio-political.

Why do we consider Italian and Spanish different languages? Because different states have arisen which have standardised different forms along a spectrum, and those have in turn been emphasised in terms of differences. Why do we call Dutch and German different languages, but then Swiss German and High German are still German? Actually I don't have the specific answer to that, I'm not a Germanicist, but I can assure you it's still basically sociopolitical.

Then why do we consider Arabic to be all Arabic? As some background real quick, on a personal note, my undergrad was in mid-East studies, and I took multiple years of Arabic both in the Middle East and my country, as well as studying Hebrew. So I can assure you that Maghrebi Arabic (Morocco, e.g.) is really quite different from Levantine Arabic (e.g. Jordan), and that in turn is distinct from Gulf Arabic (Iraq). How much each of those three can understand each other also varies. Egyptian Arabic is again different, but widely understood because well that's where a lot of the movies were made, and for this reason we also see something of an Egyptification of the dialects of places like Yemen, which would otherwise not be expected to have such features.

But what Arabic has that is Arabic is a major role as a liturgical language, which is still more or less identifiable as what Modern Standard Arabic is coming from, and in turn, Modern Standard Arabic shares a status throughout the region (albeit with minor differences) as a sort of artificially propped-up standard, in the same way that someone in Switzerland or Australia may learn in school the same sort of German being learned in Germany, even if the German they speak at home is massively different. It is this role as a major feature of the liturgical traditions of the Middle East which has tied the various countries together linguistically, when, really, one could just as easily make the case that Levantine is a language and Maghrebi is a language. But with Arabic, if you're familiar with the aḥādīth (sayings attributed to the Prophet Muhammad, which may or may not be apocryphal depending on a number of factors too numerous to get into here), there are numerous sayings praising the significance of Arabic to the religion, which in turn helps secure a place for it in daily life.

This is one major reason why Muslims will attempt to read the Qur'an in Arabic and learn Arabic in Sunday school etc, even if they are not from Arabic-speaking countries or backgrounds. There is a religious significance that has assisted the languages continued use in the region in a way that did not happen with Latin in the same way.

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Full disclosure: Not a historian of the Middle East. Just a historical linguist who previously focused on Semitic languages. I welcome others with the relevant background to hop in with additional information about any specific historical events or decisions related to the subject.