With the decline of the Roman Empire in the west, written Latin did not disappear. Instead, it became the 'international' language used for communication. Through the ages it devolved and lost its linguistic cohesion. In the fourteenth century the Italian humanist Petrarch (Petrarca) argued the centuries separating his age from the Classical Age were an era of darkness. In his eyes, the tremendous loss of knowledge could only be remedied by studying the classics, most of all the great Cicero, whose lifetime happens to overlap with Emperor Augustus' lifetime. Cicero's writings have been extremely influential, both in style and ideas. His prose was and is the model of what Latin should be like. What's being taught in school these days is the Latin as it was written in Cicero's, and thus in Augustus', time, so yes, the Emperor should be able to understand it.

If so, why didn't Latin evolve and change in a similar way to other European languages?

Latin evolved in a multitude of ways, resulting in all of the Romance Languages, which include several widely-spoken modern languages (Spanish, French, Italian, Portuguese) and many others. Here are a bunch of earlier posts

General

• Who spoke Classical Latin? How far down the social hierarchy did it extend?

• Question about the spread of Latin in the Western Roman Empire.

• Was there ever a time where Latin and it's derivatives (French, Spanish Portuguese, Italian etc.) were used at the same time?

• At what point did speakers of the various Latin dialects stop calling the language they spoke "Latin"?

• When did the Romance languages deviate so much that native speakers of one could not understand another?

Italian

• FAQ section How did Latin "die out"? How did Italian evolve from Latin?

• Could an Italian peasant (in Rome, Sardinia, Sicily, or Tuscany), circa 1100, understand much of the Latin spoken in church?

• When did the transition from Latin to Italian happen? - includes a link to a related post in /r/Linguistics

• Is the modern Italian language just Latin that morphed over a few thousand years?

Romanian

• What is Romania's connection to the Romans?

• How did the region that has developed into modern Romania develop a Romance based language and not a slavic one like Bulgaria?

• Is there an historic reason why Romanians consider themselves Latin but the bordering countries who also were part of the Roman Empire do not?

• How come Romanian is a latin language? The country was under roman domination for less than 200 years and not entirely ( around 20% of its territory).

• Why does Romania speak a Romance language but none of the rest of Eastern Europe does? which links to this quite old thread How did it end up that Romanian, a romance languge, is spoken in n area completely surrounded by slavic/hungarian, cut off from all other romance languages?

French

• FAQ section Frankish and French

If Latin were a purely written language, would it have been more stable? I know Hebrew for example didn't change very much over a similar time frame- there were some changes in grammar and some new words, but you can easily read Hebrew from any time period.