The difference between Classical Arabic (Qur'anic Arabic) and Modern Standard Arabic aren't groundshaking. Think of it as the difference between today's English and Medieval English (thee, thou, thy, etc.)
assalamu alaykum
Sorry, No.

thats far from what is correct.
Modern Arabic and Classical Arabic is very similar. Infact, Classical Arabic or Fus7a (pronounced Foos (as in Foot) Haa (as in Hamburger).. Foosha) is learnt by all the people who wish to study at a high level. It is a sign of great eloquence and education.
The normal Arabic that is spoken daily in people's homes, is called 3amiyyah. So you have slang and speech which is local dialect. Such as Lebanese, Egyptian Arabic etc.
But, in newspapers, business meetings, news broadcasts the people speak formal arabic. It is Modern Standard Arabic and is very similar to Fus7a.
Shakespeare's Medieaval English and today's English is VERY different. nobody speaks like Shakespeare anymore... However, Classical Arabic, from the time that the Quran was revealed is
more widely used and understood then any other form of Arabic.
This is one of the amazing properties of the Quran, and Allah said that he will preserve the Quran for all time.
Not only are the actual words of the Quran preserved, but also the language is preserved too. The Arabic language has not changed a great deal.
Modern Standard Arabic simply has new words that come from new inventions and concepts, such as scientific, political, journalistic words.
Written Arabic is always formal.
Nobody speaks Modern Standard Arabic at home, people revert back to the more homely tongue of their native land... so, a Lebanese person might speak and write in Formal Arabic whilst speaking more relaxed in their Lebanese Arabic at home.
Wasalam.