Why do Christian denominations exist when the Bible speaks against this?Firstly, I must point out that many people make much of the divisions that lead to denominationalism and fail to observe the overall unity of the Church of Christ. Not all "splits" were due to major disagreements over major doctrinal issues. When someone tells you there are over 40,000 denominations (as I commonly see posted on Quora), they are only giving a partial truth.
For instance, there are divisions of the church based on language and culture.
There are countries like Nigeria, for instance that report 1300 different languages (or more!), each with a distinct flavor and nuanced use of examples, analogies, and storytelling. In that nation, I know that some churches have "split" from a parent church to form a body of believers that all speak the same language and have the same cultural background. This has happened all over the world, including the USA where there are the greatest number of denominations. There was no animus in the split, just an agreement that each differently cultured portion would best benefit from a place where they could speak their own language and express their own culture as applied to their Christian faith.
Similarly, in the USA, MINOR differences that have nothing to do with major doctrine are responsible for several thousand "denominations" within the Christian church. Today, the style of music is a big issue for some churches. Where disagreement on the style of music to use--hymns from the 1800s or modern music using electric instruments and drums--leads to descension and animosity, it is quite sad and does, indeed cause a kind of disunity among believers; but it has nothing to do with major doctrinal stands or actual theological divisions in the Church. It has more to do with highlighting the "human-ness" of the people and that they still need the working of the Holy Spirit to help them better align their lives with that of Jesus.
Still other divisions are because of a uniquely American "tradition" in church government--independent church bodies. Several thousand "denominations" exist that are made up of only one or maybe two congregations of believers. These churches stand in unity of belief with millions of others, but believe that church government should be local and totally decentralized. They reject having an overseeing denominational board or a body led by a single person as in the pope. In aggregate, all these tiny churches would amount to perhaps the equivalent of a single major denomination, but they are listed as separate denominations in the books and sites that follow Christian denominations.
When you get down to the major points of orthodox Christianity; the nature of the Godhead, the divinity of Jesus, the place and inspiration of Scipture, the need for humans to rely on God and not their own efforts to have salvation, and so on (summed up in the earliest Church creeds, starting with the Council of Jerusalem soon after Jesus' death and resurrection), you'll find very few true divisions in the Church: The Roman Catholic Church (and her offshoots), the Eastern Orthodox Church (and her offshoots, based predominantly on language and ethnicity), the churches of the Protestant Reformation (whose divisions are generally related to church government, style of worship, and social issues and not divisions over fundamental doctrines), the so-called "Restoration" churches (attempting to take the Church back to 1st-Century first principals), and the myriad of Independent Christian churches which often fall under one of the above, but because of their independence from a formal governing body, refuse to be pigeon-holed under a name any more specific than "Christian."
Then we have the non-orthodox churches that do not fit into the schema above because they do not hold to the earliest of creeds (found in the Scipture's descriptions of the early Church and the epistles written to the early Church), reject one or more of the main tenets of orthodox Christianity; usually the pre-eminence of Jesus as a perfect human, his pre-existence before the Incarnation, the Incarnation and his divinity, his position in the Godhead, and his role as humankind's sole Redeemer. Other tenets often rejected include the personhood of the Holy Spirit, the inspiration of scriptures, and often, the need for all humans to receive salvation through believing on Jesus and accepting him as Lord and Savior of all.
So your questions should ask about the major differences in the the major denominationalized churches within the Body of Christ.
Now, here, we have some meat to chew on, but we should save that discussion for other Quora questions.
There are some big differences in practical theology between the major denominationalized bodies (say, Roman Catholic vs. USA Southern Baptist), but even with such seemingly large divisions, we see unity in the basic tenets as described by such creeds as the (so-called) "Apostles' Creed" or the later Nicene Creed. The divisions between these denominations are often much more about culture, residence of earthly power and governance, the centrality of the Holy Spirit and inspired scripture as the Church's guide, and about individual freedom of conscience than it is about the fundamental and central doctrines of the Church Universal.
I will agree, the separation of the Church into these main denominations and the intense divisions and enmities the separations have caused are very sad, quite regrettable (to say the least!), and distract us and hinder us from the actual work of the Church--which is to be light and salt to the world, to take care of widows and orphans, to relieve the poor, take care of the sick, lift up the downtrodden, and to proclaim the good news that is in Jesus the Christ.
Because we are all humans with human foibles. No human is perfect and we all fail at some point. That's one of hue he very things the Church was to be for: to give us a body in which we could fail and then be lifted up by others and be strengthened and improved. That still works.
However, some humans have used and still use the Church to gain personal (or ethnic, or political) power and wealth--and do not actually believe what the Church teaches. These are the wolves in sheep's clothing, as it were.
Others make mistakes and then, because of pride, refuse to own up to those mistakes and the consequences of the mistakes affect others down the road through history. Some others have made errors or have failed in some way and their church failed to lift them back to their feet and support them in love, so the person(s) rejected what they saw within their local church and projected it to the Church Universal, thus paving the way for divisions in the Church.
Yours is a good question and highlights one of these biggest issues the Church faces in our world today.