Slavic languages were written in Cyrillic, but after a while, they started to use the Latin script, and the languages were completely split into separate languages using different scripts, now being known as Eastern and Western Slavic. However, looking at Asian and Middle-Eastern languages, we see that they are not splitting into different languages using different scripts like Slavic languages. Why did that not happen to them?
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Oversimplified and generalized, but most Slavic-speaking countries and nations that use Cyrillic are traditionally predominantly Eastern Orthodox Christian, and most Slavic-speaking countries and nations that use the Latin script are traditionally Catholic. There are exceptions (like the majority-Muslim Bosniaks using Cyrillic, or the majority-Orthodox Serbians using both Latin and Cyrillic), but this usually works as a general rule of thumb.
This mostly goes back to how each group of Slavs got converted to Christianity back in the day, and how the Bible would get translated to local languages. Essentially you are looking at writing systems based on Latin created by Catholics and writing systems based on Greek (heavily modified from the start) created by Orthodox Christians. Of course, in reality things were much more complicated, but this should give you a general idea.
Also note that besides East and West Slavic, South Slavic languages are also a thing. Some of them use Cyrillic, and some use Latin!
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