This question has naturally been already brought up on many occasions, and I too have brainstormed about it often enough. By now there have accumulated numerous additional questions that one might find interesting.
1) Domestic-Territorial. The events of 1065-6 have shown that the Anglo-Saxon kingdom was very unstable; the Godwinsons and their foreing relatives kept fighting over the throne, and there was considerable regional separatism. I suppose that some degree of unification could be achieved, but as it would be harder without the Normans the Godwinson rulers might resort to even harsher measures than those of William to keep Northumbria in line. Alternatively, affairs might develop more similarily to those in Kievan Rus
sans Mongols, with internicine strife and gradual disunity, eventual economic boom (probably in non-devastated Northumbria) with resulting growth of internal trade and shift of the centre of power northwards, ultimately resulting in reunification.
2) Domestic-Political. While harder to centralise, the non-Norman Anglo-Saxon England would probably be much more absolutist (much of the OTL parliamentarism was a fruit of Norman petty feudalism and French influences, and in OTL pre-Norman Angleland was probably the most absolute monarchy in Western Europe at the time), especially after reunion. Although there was a tendency towards the rise of magnates (i.e. a few VERY powerful feudals), it is unlikely that an oligarchy could triumph in the long-term (the most powerful magnate is likely to end up taking the throne for himself). Witan is likely to lose what significance it had, over time.
3) Social. As already said, less petty nobility, more grand nobility, cities less devastated=stronger middle class, plus a fair amount of bureaucracy after the reunification.
4) Religious. Catholicism still is likely; but a more interesting and still kinda feasible option would be Saxon Orthodoxy. Divisions between Catholicism and Orthodoxy were still fairly ambigous so soon after the schism, and the Anglo-Saxons haven't been particularily known for their loyalty to the Pope; Papal support for William the Conqueror might also complicate things between Godwinsons and Rome; and the Godwinsons had Russian relatives. Besides, conciliar tendencies seem both natural for the more urban Anglo-Saxon civilisation and very compatible with Orthodox practices.
5) Economic. Likely more commerce in general, with Scandinavia in particular, though also with Rus and Holy Roman Empire. Really, not sure. Faster economic development overall, with a more intact Northumbria.
6) Geopolitical. Prior to reunification, foreign adventures would be unlikely; after the reunification, well, dynastic ties still might bring the English into French affairs, but that is unlikely. I somewhat doubt that the Anglo-Saxons would involve themselves in Scandinavia (I do think they would seize Iceland at some point), but at later stages they might try and form a coalition to counter the Hanseatic League equivalent, with other local powers like Denmark and Novgorod (both having had conflicted with Hanse in OTL). Or maybe they could cooperate with said league instead, possibly from the outset (I could easily see Anglo-Saxon and North German cities cooperate, with royal backing). In that case, it is more likely to have some conflicts with Denmark and Novgorod. The Baltic option might be a very viable and distractive one, possibly enough for England to turn a blind eye to Ireland, Scotland and Wales (possibly after grabbing the Scottish Lowlands), at least for a while. There is also the Atlantic option; it is, IMHO, both less likely and less interesting (being way overdone in althist and in OTL allike), but certainly viable. In that case the English overrun the British Isles and colonise North America. Meh.
And, ofcourse, there are the other countries. Normans might waste their potential with continued conquest attempts, or they might turn to conquering France (might work, too, or they might end up preventing the formation of an unified French nation-state; it is possible to imagine a Greater Normandy in the north while the centre and the south are in the Kingdom of Aquitaine, and Burgundy never becomes French in the first place, possibly developing its own culture), or perhaps focusing more on the Mediterranean - apart from Sicily, there is Spain, and the Byzantine Empire, and ofcourse the Holy Land. Or any combination thereof, really.
Naturally, if we do end up ruining France we will help the Holy Roman Empire, though it is unclear if this help will be consequential in the long-term.
Honestly not sure as to the effects on Scandinavia and Rus. But there will be some, I assure you.

Some ideas:
- Greater trade, as already mentioned;
- Resultingly, greater competition, and possibly trade wars;
- Anglo-Novgorodian alliance;
- To mess things up, maybe we could send some Normans to Livonia instead of Teutons, while the Germans focus more on Poland, Bohemia and Hungary in their Drang Nach Osten. Might be enough to seriously assimilate Poland after the Mongol invasion.