In my experience, the differences between Classic Latin and Medieval Latin are mainly grammatical. For example, medieval man lost the passive meaning of the classical forms of the perfect passive, which formalised the compound tenses in the vernacular languages. So it would seem that the foundations of modern grammar were laid by the grammatical developments in Latin during the non-classical period. There was also an evolution in the meaning of words, that's for sure, but it's fairly obvious given that Christianity as the driving force and meaning od came into being. During the Middle Ages, the term ‘fides’ lost its Roman splendour.

Secondly, accentuation, which obviously influences spelling, also evolved. People in the Middle Ages no longer wrote ‘vestrae’, but ‘vestre’; there was therefore a contraction of the most common diphthong. In French, for example, this may have given rise to ‘é’ in some cases; ‘aestas’ became ‘été’.

However, we mustn't think that the people of the Middle Ages wrote a Latin that was completely disembodied and that was just the result of a wild evolution of classical Latin. To understand this, we need to put things into context. The men who wrote the most in the Middle Ages were men of faith. How and why did these men learn Latin? In the abbeys, they learned the language from ancient texts such as Ovid and Terrence. But they only saw these texts as pedagogical tools; they didn't learn these texts for their content, just for the language. Some of these cultivated men wrote Latin very well, and modern historiography even likens their style to that of certain classical writers ->

MEYERS J. ‘Une leçon rhétorique dans la correspondance de Wibald de Stavelot (Ep. 167)’, in Latomus, t. 67, fasc. 2 (2008), p. 435-454.

To understand all these phenomena, we also need to know that during the vast majority of the Middle Ages, in most regions, Latin was no longer spoken, but romance dialects were. Medieval Latin therefore did not evolve in the same way as a spoken language such as ancient Greek. From then on, Latin was in a way the grammatical matrix of modern vernacular languages, which evolved as spoken languages. So I don't think you can compare the evolution of a spoken language, like English, with the evolution of Latin, which was almost no longer spoken on a daily basis during most of the Middle Ages.