The difference with formal language is not just due to Norman rulers: across the whole of Europe for almost 2000 years, the language of Law was Latin and lawyers were educated by (mostly Catholic) Latin-speaking clerics.
You can observe the same difference between "vulgar" and "formal" modes in French, Italian, Spanish and probably most other continental languages as well. 'Vulgar' would evolve quickly and absorb influences from all over (invading armies, temporary rulers, fads etc), but formal language would stick to Latin words or their closest local equivalent. This was absolutely necessary at a time when national identities and linguistic cores had not been firmly established and two neighbouring towns might be talking completely different languages for various reasons (even just out of spite for each other).
Do you know about Law French? Have you ever seen the Selden Society reports? Before 1600, the law was primarily written in Law French in the UK, a language somewhere between English and French but definitely not Latin.
You can observe the same difference between "vulgar" and "formal" modes in French, Italian, Spanish and probably most other continental languages as well. 'Vulgar' would evolve quickly and absorb influences from all over (invading armies, temporary rulers, fads etc), but formal language would stick to Latin words or their closest local equivalent. This was absolutely necessary at a time when national identities and linguistic cores had not been firmly established and two neighbouring towns might be talking completely different languages for various reasons (even just out of spite for each other).