In Arabic, the modern dialect and the classical dialect are actually quite similar, more so than ancient and modern versions of other classical languages. Absent some stylistic differences, and some basic constructions in grammar, the similarities are remarkable, and yet a little deceiving at the same time.
Of course, modern Arabic, like all other languages, contains words which do not exist in classic Arabic. Words like car, plastic, engine, etc. obviously did not exist in the ancient language. And some lexical meanings of words have also changed over time.
Like all evolution, the transition from classical to modern happened gradually. If one must define a cut-off point, one would not be too far off if one pointed to the middle of the 19th century.
Any decision about which version of Arabic to learn must be driven by the ultimate goal. On the surface, it would seem to make sense for many to study modern standard Arabic. However, if the primary goal is to study and analyze the Qur’an, then the best course of action might be to study classical Arabic.
No matter which choice one makes, it is very feasible to then extend one’s knowledge to learn the other, because of the similarities referred to above.
Either way, one should not forget that both modern standard Arabic and classical Arabic are essentially compromises. Each has variations.
In modern times, each part of the Arab world has different versions, or dialects. This means that people from different Arab regions might have mild to extreme difficulty understanding the version spoken in a different region.
The same can be said for classical Arabic. In the earliest times, each tribe had its own version, and these differed considerably. In or about the 10th century, two versions emerged as primary – Basra and Kufa. These became dominant.
For those who wish to analyze the Qur’an, learning classical Arabic is only the beginning. There is also the Hadith, and other principles of religious education, such as morality, piety, and logic. But it must start with learning Arabic, and in this case classical Arabic, unless there is also a desire to use the language for purposes over and above Qur’an study.
Whichever branch one decides to study, always remember that it is relatively easy to grasp the other, because of the large overlap and similarity.