How would a surviving Anglo-Saxon dynasty affects England and British Isle in all aspects: politics, culture, science and tech, economy, military...?
OTOH, England would avoid significant manpower and wealth losses due to the fact that the Normans exploited its wealth and devastated the North when quelling rebellions. Also, the Hundred Years' War might have been averted.
OTOH, Norman rule brought certain long-term benefits.
The biggest single change, administratively, was the redistribution of land. A bunch of Anglo-Saxon landowners were dispossessed (rather quickly it appears; some of them probably would have died at Hastings) and their land went to the families of those lords that had helped William secure the place. The fact that the King could now grant land as a reward to his loyal followers had a massive influence in his ability to raise armies, levy taxes and gain the co-operation of local lords. In Anglo-Saxon England, land belonged inalienably to individual landholders, which if you are trying to develop a powerful government is a real hindrance.
Not to say that Anglo-Saxon England wouldn't have centralised without the conquest: basically all Western European states did at one point or another. But some were much faster at centralising than others, and that certainly helped England in its competition with other European states. Medieval France and the Holy Roman Empire are both good examples of states that were plagued by a lack of central control, much to the frustration of their monarchs' ambitions. In the case of France in particular, it suffered during the Hundred Years' War because England was basically more cohesive and more unified, with a somewhat more developed sense of national, rather than regional, identity. Centralised government also helped the English to dominate their neighbours, the Welsh, Scots, and Irish. An Anglo Saxon England might even butterfly away United Kingdom.
Besides, the Norman introduced cavalry warfare and stone fortifications and structures (like cathedrals), all of which were rare in England before 1066. Not to mention that archery was rare in Anglo-Saxon England. The economy expanded as port-building (for example
Boston, Lincolnshire) took effect. The army and navy were strengthened.
Finally, Norman kings also joined crusades. The crusades increased England's exposure to the cultural and economic centres of Medieval Europe, such as Italian city-states, Byzantine Empire, and Muslim world, and hence learned lots of valuable scientific and technical knowledge, as well as military tactics. While Medieval England was quite prosperous by Western European standards, it was a complete backwater compared to these regions. Meanwhile, an Anglo-Saxon England would be less involved in these affairs, and would focus more on Scandinavia, which was also a backwater compared to Southern Europe or Middle East. Later, the Hundred Years' War revolutionized warfare with the introduction of gunpowder and firearms.
So, an Anglo-Saxon England would be very likely to lag behind the rest of Europe in construction, science and technology, as well as military.