As the other posters told, the thing is that the Bible and the Qu'ran are very different in nature. The first is a collection of many different texts written (and modified) by many different anonymous people over hundreds if not thousands of years. It has always had many different versions, in fact the New Testament is precisely a collection of different versions grouped together. Meanwhile the Qu'ran has only one author, who allegedly received it directly from God and made people remember it verbatim, word for word, with order never to change anything from it. Obviously in those conditions it's normal for the Qu'ran to have a single version, just as we have essentially one version of Plato's Republic or of Virgil's Eneid.

Compare that to the NT, which is not a single book written by Jesus, but a compilation of memories attributed to his disciples and which was written at least decades after his death. In Islam, the equivalent would be the Hadiths, records about the life of Mohammed, which were also compiled long after his death and are equally contested, with different groups accusing each other of forging fake hadiths, and recognizing different ones as authentic.

But we should also add that there were different versions of the Qu'ran during the first decades, because it was learnt by heart with very few written records. It requires incredible memory to remember such a long text entirely and perfectly by heart, so of course soon enough there were little variations. As /u/IamArabAndIKnowIt said, the third Caliph ordered to destroy all versions of the Qu'ran but one, which would be the new, single, standard one. We don't know for sure but allegedly the points of disagreement between the different versions were very small and didn't really change much. For instance in the standard version it talks about "the middle prayer" and in another it talked about "the afternoon prayer", but the afternoon prayer is the middle one (3rd on 5). There are also claims about "lost verses/chapters" but once again it's difficult to tell, and as far as I know noone uses a different Qu'ran including them.

>Do we know if different Islamic sects had different Qu'rans?

I can answer this question from real life experience, and can back it up with sources if you so deasire, but, yes there are sects within Islam that do not recognize the Quran as the shia/sunni interpret it, or even go so far as do covertly deny the authority of the quran.

One such sect is the one I adhere to, called the Alevism, where the belief is the quran as it is today has been manipulated and re-written over and over again to suit the needs of those in power (mainly the 3 Caliphs before Ali and the Umayyads).

Rather than the Quran, the Alevi have the buyruks as their authoritative body.

It might be pertinent to point out that in Shia Islam the Imams were regarded, while they were on Earth, as the interpreters of the Quran, whom could do no wrong, thus had the same, if not higher, authority than the quran. Living saints if you will.

When comparing to the scale of the Bible, you'd say that there were almost no versions of the Quran. One perspective has to be put under the spotlight when it comes to this issue. The writer(s), and the time of writing after the origins.

The Bible has so many authors, and until this day, most of them are unknown, even the ones that are said to be written by X, Y, and Z. Even almost half of Paul's letters. See Bart Ehrman's books for more details. Above that, the Bible was collected after more than 300 years of the original stories/events.

On the other hand, the time era of the Quran has happened during a time that was way more advanced compared to 600 years earlier. And also, it is claimed to have had only one author, God, or Muhammed!

The single author of the Quran made it a lot easier to not have self contradictions (or a lot, relatively). But with the Bible, stories have gotten way too adjusted and modified, so it's very easy to find some bizarre self-contradictions.

When the Quran was written though, it was mainly for Arabs in the Arabian Peninsula, and it was readable for all of them. But Islam has spread too fast beyond that area, and non-Arabs started to try to read the Arab writing (which by that time was without vowels, and the natives just figured out what was written based on just the key sounds.)... That confuzed the crap out of non-Arabs, and they started reading the Quran in all different ways. That some times resulted in screwed up meanings! This made the third Chalif of Islam decide to remove every Quran writing and start from scratch! This is why he burned everything and decided to modify the Arabic writing style. This made the book very united until today!

Here is a picture comparing the old vowel-less writing of the Quran and the modified style:

https://29letters.files.wordpress.com/2007/05/arabic-letterforms.jpg