The Anglo-Saxons practiced Christianity and accepted the authority of the Roman Catholic Church, yet the Pope gave his permission to William the Bastard to invade England to "advance the Christian religion." What practices in the Anglo-Saxon Church caused them to gain the scorn of the Pope?
So this is a little difficult to fully tease out. Anglo-Saxon England had been Christian for a long time when William pressed his claim to the throne. It was converted over the course of the 7th century and had ceased to have pagan realms by the end of the 600's. Following the Norse incursions of the 9th-11th centuries Christianity was firmly in control of the British isles with England in particular producing a great deal of the clergy that was used in converting Scandinavia, much to the chagrin of the Archbishops of Hamburg-Bremen.
However England had a history of some difficulty with the Roman Curia at this point in history. Despite his late canonization, Edward the Confessor seems to have turned a blind eye to many abuses of ecclesiastical norms in his realm. For example, there was a bit of a tradition of bishops in England holding multiple sees simultaneously, which was not looked kindly upon by the Papacy, being connected to simony, or the selling of Church offices and practices. This was indeed a source of consternation between the two for a while. Individual English kings did undertake pilgrimages to Rome and wrestled some concessions on the payment of tithes and dues for bishops taking their palliums back to England (this was attributed to Canute the Great), but there were other tensions. England was also a hotbed of the slave trade at this point in the Middle Ages, and the practice was supported and patronized by the English Church. Slavery at this point in the Middle Ages was unpopular in western Europe, and enslaving fellow Christians was also quite frowned upon. Indeed, slavery would not be removed from England until after the Norman Conquest. Other issues such as clerical marriages.
However there are a few things to keep in mind as well, while papal support for William's invasion is a part of post-Conquest narratives, we need to keep in mind that the Normans had a vested interested in playing up the legitimacy of their intervention into English affairs. The movement to canonize Edward the Confessor, claiming that Harold Godwinson had consented to William's claim to the throne (sworn upon holy relics), and so on all play into this. So while there is little reason to doubt Papal support of William in some form, we should probably not assume too much in this regard.
I also wrote an answer to the same question for a while before, though I afraid my answer(s) had been not so persuasive as /u/Steelcan909 does in this sub-reddit.
I personally have some doubt on the authenticity of this claim, i.e. whether the claim could be fabricated retrospectively from ca. 1070 when the relationship between new King William and Anglo-Saxon archbishop Stigand did not go well clearly.
(Added): The following three books are classics for your question's topic, I suppose:
The Anglo-Saxons allowed the bishopric of Canterbury to go a supporter of Anti-pope Benedict X. This obviously led Pope Alexander II to support William's claims. Even more importantly, Alexander II's inner circle (especially Ildebrando, the future reformer Pope Gergory VII) was convinced that the English church strongly needed Reform and drastic change (to fight simoniacs and stuff like that). William will eventually reform the English church and have three papal legates depose the Bishop of Canterbury.
C. Morton, Pope Alexander II and the Norman Conquest,