In addition to the all the good reasons cited in the previous answers, I'd like to emphasise the role of the Catholic Church.
When (ca. 496) Clovis, then young king of the Franks, resolved to convert to Catholicism, allegedly under the influence of his wife and Saint Remigius but more probably because he understood what a fruitful collaboration he would inaugurate, he implicitly renounced to impose Old Frankish as the new official language of the Diocese of Galliae (later known as France).
As a result of his own conversion, along with that of his whole army, he secured the support of the Gallic Catholic Church in his fight against the other Germanic rulers who had their eyes set on Gaul, as well as the loyalty of a large number of learned and devoted administrators only too happy to serve him.
These newly appointed administrators, the rest of the clergy, and in their wake the cities and the hinterland just stuck to Late Latin.
There would have been little chance anyway that the invaders could have been able to impose Old Frankish to a Gallo-Roman population 5 million strong as they were counting for less than 5% of its total. They probably did not make any serious attempt to do so anyway but were instead keen to step into the shoes of the incumbent Latin speaking ruling class. What would later become known as France had been Romanised during five centuries (the time separating us from the Tudors) and Gaulish had long been extinct.
A similar phenomenon took place at exactly the same time in Ostrogothic Italy where Theodoric - earlier brought up as a hostage in Constantinople - exerted power surrounded and advised by learned scholars speaking both Greek and Latin as well as in Visigothic Spain where Theodoric was also a regent. In all cases the Germanic ruling class was not only a linguistic minority but also keen to win the hearts and souls of the prestigious local Latin speaking elite.
The prestige factor was actually determinant as well. On the continent, Germanic peoples looked up to the Roman Empire and had no intention to ruin it. Their rulers in Gaul and Italy were eager to slip into the imperial trabea1.
The situation of Roman Britain at the turn of the fifth century stands in stark contrast:
- Its Romanisation was only effective in the cities (esp. the Midlands and London) - most of them garrisons (70 places end in -chester, -cester, -castre and -eter).
- Various Celtic languages were still the main tongue of a large proportion of the low classes.
- The borders with Wales and Scotland were marked by intermittent unrest.
- The proportion of the population of genuine Latin ancestry was fractional.
- The total population the Anglo-Saxon had to subdue was around one million2.
- Above all Late Roman Britain was marked by a revival of paganism3 so that there was no reason for the Anglo Saxon tribes to look for an alliance with a Catholic Church which had little to offer.
- Finally, the occupation had predominantly been a military one. Once the legions had reembarked, the invaded populations looked more like uncouth Celt peasants than polished Roman citizens. The prestige factor thus actually went in the opposite direction: the Celtic upper class soon integrated with the Saxon invaders.
I realise that I've actually explained why Romance Languages survived in the few places where they did but if one takes a broader view, there are actually more parts of the former Roman Empire which did not retain a Romance language. In North Africa and in the Balkans for instance. Not to mention the former Byzantine Empire where Greek was the official language anyway.
: In this respect, it is worth remembering that the supreme title Kaiser
(and Czar/Tsar) so proudly born by the descendants of Ariovistus
until 1918 (Kaiser William II) is the very cognomen of the Roman General who defeated him: Julius Caesar. And that the phrase "Holy Roman
Empire" designating the German Empire lasted till the beginning of the 19th century.
Note 2: Estimations of 2 millions seem exaggerated - 1/2 million in 650AD.
Note 3: See "Religion In Late Roman Britain" - (Dorothy Watts 1998 Routledge) in particular Chapter 2 "The Revival of Paganism of the Late Fourth Century". [This is just before the evacuation of Britain by the Roman legions].