It's a bit early for my period so someone with more expertise could come along and provide more information but I can do a basic rundown.

First of all, it should be said that your question sets up a false dichotomy. The Bayeux Tapestry is a hugely important historical source...but it's not the primary account of the events it depicts. 11th century medieval European art is really rare so for that alone it's a big deal. Its existence is almost as fascinating as the story it tells since it's such a weird piece of art, it's not really a tapestry since it's embroidered rather than woven, and we're not really sure how it was displayed. It's not an easy piece of art to display due to its great length.

I wrote a response a while ago on Harold's death that discussed the Bayeux Tapestry and other sources that talked about the battle of Hastings. You can read the full response here but I'll run through the main sources for William's English conquest below. I should also note that the Bayeux Tapestry actually includes some other campaigns of William (to Brittany I believe, but I'd have to double check that). I'm not sure if these sources include accounts of the Brittany campaign since I've basically never studied it.

The various chronicle authors who are generally discussed when we talk about 'reliable' (for medieval!) accounts of the Norman conquest are:

William of Poitiers (c.1020-1090) was a chaplain to William the Conqueror and wrote a history of William's conquest for him. Often considered the primary Norman account of the conquest.

William of Jumièges was a contemporary of the battle of Hastings but we don't know very much about him at all besides that. He was a Norman monk born probably sometime around 1000 and dying sometime around 1070. He wrote an account of the deeds of the Dukes of Normandy that includes as part of it William's conquest of England.

The Anglo-Saxon Chronicle is pretty much what it says on the tin. As a work it doesn't have a single author. It was originally a collection of annals bound together around the 9th century but in a very common medieval trend many authors updated copies of that afterward. There are several versions as a result as different authors wrote continuations of the copy they had. At least one includes details about the Norman invasion.

Henry of Huntingdon (c.1088-1157) was a bit of a later source but drew on many earlier accounts. He's also pretty close to contemporary for a medieval source.

William of Malmesbury (c.1096-1143) is later source but also held in high regard by a lot of modern historians.

All of these chronicles allow us to piece together the narrative of the Norman invasion. While the Bayeux Tapestry is a hugely valuable historical source it is not the one most scholars would go to first when making an argument about the events of the Norman Conquest. Usually arguments about what is shown in the Bayeux Tapestry use something from one of these chronicles to supplement the point. This is in part the preference for written materials common to most historical study and in part due to the fact that the Bayeux Tapestry is not really that easy to read. The debate around Harold's death is the most notorious case of the tapestry being unclear but it's not alone.