Some people have mentioned that the Arabic language is indeed dialects while Latin has spawned entire languages, but I think this is not a fair representation of the reality, and misses a few interesting points.
Sure, Latin has given birth to many different languages, and Arabic dialects, but the difference, from the question's point of view, is that one is still used and, for the most part, understood. An educated speaker of French or Italian will not understand Latin being spoken, as it was spoken many many years ago. However, an educated Arab should be able to easily understand classical Arabic to a degree and MSA completely. Now, with that distinction out of the way, I will answer the question as best I can. Religion. Arabic and Latin are both languages used for religious purposes. However, Latin became a religious tool of the Christian church at a point where it was not the only source of literature of vast importance. However, Arabic become a big thing directly because of Islam. The views of Islam see Arabic as a highly important part of the religion. Islam has largely persevered Arabic and maintained it. Latin did this for some time, but differently. Ordinary people were not encouraged to learn it in any fashion, and ignorance in it was a tool for the higher ups in society (sorry to get conspiratle). So, I think this answers the question in a sensible manner.