Oh, if I did not have a midterm on Veda and Torah right now, I would sooooo be able to write a sweet answer to this question!
However, just a few quick and probably imprecise thoughts that I wanted to make on reading the early responses before I come back and make the big scholarly answer that does not actually answer anything but looks good on my future application for tenure…
The schism happened for lots of reasons, most of them having to do with politics. As the Papacy developed and began to be much more independent of any Imperial influence, the Roman Emperor having long since moved out East, the Papacy began to develop more its Latin character and assert its authority, all Churches agreeing that the seat of Peter held some sort of nebulous primacy, differing to the extent depending on whom and when you ask. Bishops tended to be jealous of their authority, as is natural.
In the meantime, Constantinople, as the new capital, began, both by its own desires and through the desires of various Emperors, to seek more power, claiming to be the “New Rome.” As you can imagine, that sparked a lot of arguments. As Muslim powers took over many formerly Byzantine lands, and simultaneously the Byzantine Empire had something of a Golden Age, formerly higher ranked Sees like Alexandria (the traditional second See, after Rome) and Antioch lost a lot of influence and power in the East in favor of Constantinople. The two Big Sees kept butting heads, and, eventually, had a bunch of splits and reunifications (which generally did not affect the common people as much as the whole Crusades invading armies thing did), not permanently splitting off until Ottomans took Constantinople and killed a bunch of people and reinstalled new Church leaders who were anti-Roman. This, I believe, led to the main reason why the split is still around today, and why I think that reunification is likely once the generation that grew up under Communism dies off. Of course, there was also general Russians being Russians that did not help, and Rome was hardly blameless of arrogance or a lack of tack.
Umm, yeah, so, that is my nutshell history, that I advise you take with a big grain of salt because I have hinduism on my brain and no sources on hand. I’ll come back and expand on it later, promise ^^
The point is, that the lineages did not really split off for doctrinal reasons, in my view, and that the doctrinal disputes that the East tends to have is a byproduct of the geopolitical situation. It doesn’t really matter which crowd you follow, both have relatively ok lineages. So, from the Catholic perspective, the split is really just a superfluous tragedy that will one day be corrected when people get their heads out of their a****, 1000 years at least too late.
The Oriental Orthodox represent a whole other cool history that I will neglect for now, even though you should look into it if you like that sort of thing.
Other thoughts: To say that Christianity is an offshoot of Judaism reminds me of the funny debate in the very early Common Era about Esau and Jacob. Both sides agreed that that metaphor applied to their situation, but each assigned to the other the role of Esau, the elder (and bad one). The point is, that at the time Judaism was changing a lot, and I believe the proper and consensus view among religious historians is to view them as sister faiths rather than one being older than the other.
On what the original Jewish Christians taught - Ummm, try reading Clement, for a view on non-Pauline Christianity, such as it was. Far from complete, though. Haha, I really wish I was not busy today…
This is a good book, if a bit controversial, about the split between Judaism and Christianity.