Going on the lecture given by Professor Freedman in the Yale Open Course "the Early Middle Ages", the Roman cultural influence did not survive the abandonment by the Romans in the same way it did on the mainland. The Anglo-saxons, who proceeded the Romans, conquered much of England, except for Cornwall and Wales. I suspect the survival of celtic culture has more to do with anglo-saxons than the Romans. See here for the video. He discusses the relevant topic at around 5.00

I apologize for the late answer, just stumbled across this thread. /u/Keito's link provides a basic reason for the continuance of the Celtic culture in Wales and Cornwall, but does not go into the why. The reasons are economical, geographical, and religious.

Prior to the Claudian invasion of Britain in 43 AD, the tribes of southern Britain were divided economically and technologically. The tribes of southeast Britain traded extensively with Rome long before the conquest and possessed Late Iron Age technology, while those in the southwest Britain, the regions of Wales and Cornwall, were dependent on their neighbors for continental goods, and often still had earlier Iron Age technology.

This situation itself came about as a result of Caesar's campaign against the Celtic Venetii tribe in 56 BC. The tribe, from the modern area of Britanny, France, was annihilated by Caesar and sold into slavery, but before this, they were one of Britain's strongest trading partners, using their fleet to shepherd goods back and forth across the Channel and acting as economic gatekeeper to the isle. More importantly, they traded heavily with the tribes in west of Britain, the Silures, Ordovices, Dumnonii of Cornwall and Wales. With the Venetii gone, Roman merchants quickly replaced them, favoring trade with the southeast, whose geography was more familiar. As a result, the western tribes were economically devastated. This likely led to a deep resentment against the Romans, if the bitterness of their later resistance was any indication.

So, it is of no surprise that when the initial resistance to the Roman conquest in 43 AD, led by the Catuvellauni ruler, Caratacus, failed, he fled to the Ordovices and Silures of Wales to rally further resistance. And resist he did, until 50 or 51 AD. Ultimately, it was betrayal by the Queen of the Brigantes, Cartimandua, that delivered him to the Romans. Even then, Tacitus said of him: "His fame had spread thence, and traveled to the neighbouring islands and provinces, and was actually celebrated in Italy" (Agricola 12.36). He was eventually pardoned by Claudius if Tacitus was correct and lived out his life in Rome.

The Silures and Ordovices resisted still though, supposedly in Caratacus' name. A number of cohorts and nearly and entire legion was lost in the effort to pacify the hills of Wales. Suetonius Paulinus' massacre of the Druidic stronghold at the Isle of Anglesey enabled the success of Boudica's rebellion in 60-1. Caesar himself complained of the Druidic religion and how it fostered resistance to Roman rule in his De Bello Gallico. The fact that a religious center was based in Wales undoubtedly led to the longevity and nature of the resistance, and the preservation of the Celtic culture in that region.

In the end, it would be Julius Frontinus, one of Rome's more brilliant military minds, that destroyed the Ordovices during his governership of Britain in 72 AD, wiping the tribe out completely according to Tacitus. He doesn't even record any of them being sold into slavery. Keep that in mind, the Romans literally had to kill every man, woman, and child to end the Ordovician resistance. The Silures fought on longer still until pacified during the governorship of Julius Agricola.

The creation of the Welsh and Cornish languages is an indication of how bitterly the residents of the area resented Romanization. Where elsewhere the Celtic language faded, in those areas it continued and evolved. In Wales, the culture was preserved orally by "bards," as the Druids had always said should be done for hundreds of years, and come to us in the form of Welsh poetry (The Triads) and prose (The Four Branches of the Mabinogion) that was finally recorded in the 9th century. The origins of the Arthurian legend are found in these works as well, not unexpectedly. It is, after all, the story of a leader defending his homeland against invasion, much akin to Caratacus'. In fact, Caratacus is the latinized version of the name, in the original Celtic it is Caradog or Caradoc, and there are a number of figures in Welsh mythology with such a name with heroic traits.

This post has gone on much longer than intended, but to form a synopsis. The geography of the land meant the Romans were not as interested in subduing it as other regions. The tribes of the area had a lot of deep resentment to the Romans for prior transgressions, and in the area of Wales was one of the key Celtic religious centers that pushed for resistance to Romanization. The culture was preserved orally, in the manner of the Druids, for hundreds of years in the form of Welsh/Celtic poetry and prose.

Sources: Tacitus, The Life of Julius Agricola

Rachel Bromwich, Trioedd Ynys Prydein: the Welsh Triads of Britain (preservation of culture, sources, etc.)

Sioned Davies, The Mabingion

Julius Caesar, The Conquest of Gaul (I used the penguin edition)

Cassius Dio, Roman History (For some information on Caratacus)

Peter Salway, Roman Britain (He discusses the segmentation of tribes)

Graham Webster, The Roman Invasion of Britain and Rome Against Caratacus (segmentation of tribes, invasion, and aftermath)

Sheppard Frere, Britannia (general information)

Barry Cunliffe, Late Iron Age Communities in Britain (Info on tribes and culture)