This is a surprisingly difficult question. It's kind of the entirety of what historiography and historical methodology is about, and your questions are the ones historians have wrestled with, and have found no definitive answer (nor do I think one ever will be found).

If I were to boil it down simply, I would say history research is about these linked things: convention, information, and compromise.

Convention because styles in ways of thinking roll in and out. The way people thought about a subject like "end of civilization" 100 years ago would not be the way they think about them today. There may, in fact, be nothing wrong about the way people in the past thought, its just that it's "out-of-date." Peter Heather and Bryan Ward-Perkins would be perfect examples of this with regards to late antiquity. They're trying to bring back formerly out-of-fashion ideas regarding migration and economic collapse combined with new evidence to support these new/old theses.

Now, the common criticism is not that they're wrong. Their scholarship and their logic, is sound. But the criticism is more "you're answering an old question with new evidence, but can't we ask newer questions?" To put a spin on a more blunt reddit saying, it's like saying "you're not wrong, you're just boring."

As /u/tiako mentioned in a post a long time ago, and an idea I've taken to heart, the purpose of a thesis should not be to "select" the evidence/information to prove your idea, but to "unify" the evidence. Anyone can be selective. Holocaust deniers are selective. The Tea Party is selective. What research should attempt to do, is look at the volume of information out there, and unify as much, and as best as possible, and to not discard without good reason.

And this is where compromise comes in. Compromise is how convention selects from unity. How "good reason" matches up with the "unity of information," with a basis in learning from the trails already blazed. Information is always growing, and with it comes the need to find new ways to organize it. History is both style and substance, the possible and the "real". There is not one universal truth (though we all really really wish there was one), there is only the attempt to find the truth.

What determines what information we use, is this north star of compromise. But one should keep in mind, because both information and convention are ever changing, the answers we can answer with the information/styles we have now, ultimately are only relevant to us at this particular moment. But so long as we embrace the compromise of convention and information, we will keep history from becoming what we really don't want it to be, and that's propaganda.

What u/bitparity said, but adding a bit. This is a very difficult question to answer and no answer is really short. First, there’s historiography/historical method. Like how philosophy has different schools of thought or art has different movements, so does history. In fact, a lot of these schools are tied to various philosophical schools or different disciplines. It all depends on when, were, and what one is studying, which is part of the beauty of history. For example, Marxist history emphasizes social class and economics, the Annales school focuses on social history, feminist history focuses on women, and so on. No one school or style is right, even if some would have you think otherwise. Looking at things from different angles is good, even if it means considering two completely opposite viewpoints.

History is a science which means, just like a chemist running through a set of steps in a lab experiment, historians are running through a set of steps when looking at sources. Every source has a bias or is unreliable in some way, even something like a census record that appears to just be no-nonsense tick marks in boxes. Part of a historian’s job is to consider what biases a source might have. Who created the source, when/where was the source made, who was the audience, why was the source created, etc. There will be sources about the same event that say completely different things. There will be people who lie, knowingly or not. Things will get lost. It’s weighing each source individually and determining how reliable it is and recognizing that we can only try our best to get as close to the truth as possible. Stuff also gets found all the time too. When new evidence appears, whether or not it contradicts the most accepted theory, you don’t ignore it. That’s bad science.

Historians do disagree and that’s very important. Again, that chemist in a lab, running experiments has to let their experiment be tested by other chemists. Historians peer review other historians and make sure they "show their work." It's about discussion and exchanging ideas, not proving someone wrong/right. History in schools is a bit tricky. Certain schools might use certain books that subscribe to only one viewpoint to further their own agenda. Text books are generally watered down history to begin with (at least in lower grades). It's getting better, but schools still usually just teach the most agreed upon theory.

Now as to why anyone would/should care about history and how it impacts today is again a very difficult and lengthy answer. As much as I hate the phrase that "history repeats itself," there is something to it. Some things that happened in history and things that happen today are very similar. An easy example would be the repeated invasion attempts of Russia, although there hasn't been one recently to my knowledge. It wasn't successful for Germany, France, Japan, the U.S., The U.K., the Swedes, and many many others so one would think someone would've taken the hint. But you're (probably) not going to invade Russia, so why should you care, especially about something like a [Catholic festival practiced by Italian Immigrants in Harlem in the early 20th century?] (http://www.amazon.com/The-Madonna-115th-Street-Community/dp/0300042647) It doesn't drastically hinder your life if you don't. However, it might be useful to understand a piece of urban/social/religious/immigrant history that makes Harlem as we know it today. You should care because it effects the world you live in, but in the end only you can decide how much you care.

edit: spelling.

Specifically addressing Columbus, he is certainly a magnet for debate that is much politicized. There is so much pop history material out there, from basic elementary school history books to a popular web coming from the Oatmeal. And through history he has been used as a mascot of various political movements, from the celebration of Columbus Day to the Knights of Columbus to an object of criticism and condemnation. So there is a lot of emotion involved when he is discussed.

On the other hand, there is an obsession to look for a "smoking gun" evidence and historical determinism, plus trivia-seeking tendencies.

Take as example the rediscovery of the report of Francisco Bobadilla, which was widely reported and then taken to be the proof of Columbus' cruelty, which tended to be written as the only proof of Columbus' cruelty. This is absolutely not the case. His own diaries record many of his misdeeds, and Bobadilla was sent to the colonies based on a large number of reports and complains of cruelty.

But both defenders and detractors alike flock to the Bobadilla report, which at the end of the day was immaterial because (1) Isabella and Ferdinand never prosecuted Columbus, although they never appointed administrator again, and (2) Bodabilla perished and was unable to challenge that reversal. Of course, many still think that "Gee, if I can disprove Bobadilla's report then I could prove Columbus wasn't cruel" and I expect we will still have many many questions on this subject.

sigh, this isn't even really my flair area

Now add to all this those who claim "Columbus wasn't any more cruel than other people of his era. Look for example the Spanish Inquisition." Well, this is patently incorrect because the Spanish Inquisition has specific rules against torture and specific rules about due process.

What you asked about

Do historians disagree on things, if so why?/what distinguishes these "historical facts" from those more widely accepted and taught in schools?

this is a great question, and thus the importance of clearly stating sources, such that we may revisit them and examine them in the future.

A great example is in W. H. Prescott, History of Ferdinand and Isabella where addressing the claim that Joanna the Mad had kept the corpse of her husband for months, he wrote in a footnote where that claim first appeared and stated that it could uniquely be traced to a contemporaneous history of Spain written by Peter Martyr, who himself was not present. Now, is it a wonder that Prescott is often credited to be North America's first scientific historian?