At which point in history was the language spoken not anymore called Latin but any of the succeeding languages like Italian, French or Spanish? What are the characteristics which made them different languages?
Some of the reasons
- after the fall of the Western Roman Empire, contact between various regions slowed down and lost a lot of its importance; thus the strongest reason to maintain the unity of the language ceased to work
- also, at the same time Roman bureaucracy ceased to require correct Latin
- when there are no strong reasons to keep a language united, regional dialects diverge with time, this is natural, but would probably take slightly longer to make those dialects full-fledged new languages
- various Germanic tribes with their own languages/ dialects of Germanic language (or something in between, I'm not sure how different they were) ruled various parts of the former Roman Empire, influencing Latin spoken there in slightly different manners
- the Iberic or Celtic languages spoken there before arrival of Latin were not official, but somewhere they survived in the countryside (definitely in Brittany) - this probably wasn't as big influence as we might expect in the age of "celtomania", but it was an influence
I'm just an archaeologist, not linguist, so I'll leave the second part of your question (what made them different) to some linguist, but it happened in Early Middle Ages, in few centuries after the fall of the Western Roman Empire.