Old English instead of Latin in early Britain

For almost 400 years, Britain was a Roman province. During that period, naturally, Latin was an important language in the region. When the Germanic tribes invaded the British Isles (around the 5th century), they brought with them several Germanic dialects that ended up becoming the predominant language, known today as Old English.

At the same time, other Roman provinces, such as what we now call France, Spain and Portugal, were also being invaded by other Germanic tribes. But, instead of imposing their own languages, the Barbarians themselves became "Romanized" and began speaking the local Latin dialects. If that process had also happened in Britain, they would probably be speaking a Romance language today. So, why did it not happen? How could the Anglo-Saxons manage to displace the original language spoken by their conquered people?

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.

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].

First of all, Roman presence in Britain lasted barely three hundred years. And some Latin survives, as has been noted elsewhere on this board, in place names.

Second, the Anglo-Saxons didn't totally displace the languages they encountered. Instead, they absorbed much that they encountered. For example, John McWhorter asserts in Our Magnificent Bastard Tongue that the "meaningless do" ("Do you have any money?" "Do you like fast cars?") and indicating present tense with the progressive form ("I am typing on a keyboard.") are artifacts from Cornish and Welsh. The Danes also had a huge influence.

just my 2 common cents:
It's an island. While Barbarians on the mainland would typically be only in transit, even if they 'conquered' something, the situation in Britain was a bit different, the conquerors would cohabit with native population and cultural imprint was easier.

Also, on continent the lands of France, Spain and Portugal were still close to existing remains of the Roman empire for quite a few centuries.

See a timeline here.