Every time I have a space between them my answer or question gets edited. This answer suggests it's not a big deal.
The answer you link to is misleading. It reads:
another = an + other.
The equals sign almost makes it look like the two are interchangeable, but they are not. I might have been tempted to write it more like this:
an + other → another
to show a one-way transformation (as opposed to a reflexive operation), which would be more accurate.
There are sites on the web that explain this:
At the conclusion of a long explanation about the word another, Woodward English mentions:
Remember another is ONE word not two words (an other is incorrect)
Another website clearly states:
When we use the indefinite article an before other, we write it as one word: another.
This same site goes on to reemphasize (under a header called "typical errors"):
• We write another as one word:
There is another car park a little further down the same street.
There is an other car park...
And this bit of information may be of interest, too:
“Another” was originally two words, according to the Oxford English Dictionary: “an other (often a nother).”
Until long into the 17th century, “nother” was a common pronoun and adjective meaning “a second or other; a different one,” the OED says. This use of “nother,” according to the OED, is largely obsolete today and survives only as a colloquialism in the United States, where it’s commonly used with “whole.” [emphasis added]
But that's a whole nother story... :^)
In The Cambridge Grammar of the English Language, the word another is described on page 391:
Determinative another derives historically from the compounding of the indefinite article and the adjective other; the consequence of this for the modern language is that the existence of the determinative another blocks the co-occurrence of the indefinite article and other as separate syntactic constituents: *an other book. Determinatives other than the indefinite article precede other without such compounding [ . . . ]
You can combine most determiners with the adjective other freely:
the other book ← OK; the is a determinative in determiner function
my other book ← OK; my is a genitive pronoun in determiner function
But you can't combine an and other this way:
*an other book ← ungrammatical
another book ← OK; another is a determinative in determiner function
Is this just spelling? No:
the three books ← OK
the three other books ← OK
the other three books ← OK
*a three books ← ungrammatical
*a three other books ← ungrammatical
*an other three books ← ungrammatical
another three books ← OK
As you can see, another is not simply the combination of an other. It has its own patterns of usage unlike any other word in the English language and requires its own grammatical description.
Importantly, it's a single word in speech, too. We normally connect words in speech, and if an other were grammatical, it would usually be pronounced the same way as another:
an other /əˈnʌðər/ (connected speech)
another /əˈnʌðər/ (connected speech)
So in normal speech, we have no indication of whether it's one word or two.
But speakers can optionally pronounce words separately, if they like. For example, the other can be pronounced as two separate words with a pause in between, using the strong form of each word:
the other /ˈðiː ˈʌðər/ (pronounced separately, each word emphasized)
the other /ðiˈʌðər/ (pronounced together)
But this doesn't work with another. The strong form of an is not available, and the /n/ cannot be separated from the second syllable:
an other /ˈæn ˈʌðər/ ← not OK
another /əˈnʌðər/ ← OK
As you can see, this word is not only inseparable in writing, but in speech as well. This gives us a solid argument that another should be considered a single word, not just an arbitrary spelling of two words.
In fact, many native speakers are unaware that another used to be made of an + other. Because the /n/ is always pronounced as the onset of the second syllable, some speakers have even re-split the word as a nother, as in the colloquial American idiom a whole nother.
Similar things have happened in the past. A napron became an apron, and now everyone knows the word as apron. Few speakers today have ever heard the word napron. Would it be logical to say something like this?
Sure, it was true historically, but it doesn't make any sense when you consider the modern language by itself. The same is true of the equation you found:
It's true that there are some combinations where the two words an other appear in sequence. This happens, for example, when other doesn't directly modify the head noun, but appears as part of a larger phrase:
an [other than honorable] discharge ← OK
Here, an is followed by a phrase which contains other. That phrase does include the word other, but that's okay. Cases like this are possible, but they're quite rare.
It's also okay if other is being used as a noun, although this doesn't happen very much:
The psychiatrist is emotionally and spiritually dead. He has no purpose or cause to give his life significance. One of T. S. Eliot’s “hollow men,” he lacks enthusiasm for his vocation, not believing that he has been called by an Other. (source)
Quotes like these are okay too. So how can we formulate a general rule that allows "exceptions" like these but disallows *an other? Something like this:
The sequence *an other is ungrammatical when an is in determiner function and other is an adjective in attributive function. The single word another is generally used instead.
"An other" has a different meaning from "another". As others have said, you will use it when you are using other as a noun (fairly common in philosophy), so:
As flesh of the world, I am (like all human beings) open to otherness; I am distant to myself; I am an other and an other is my self.
You can also use it as an adjective to mean "a different", for example in this book title:
Data Reduction of Wake Flow Measurements with Injection of an Other Gas
As you can see, the usage is uncommon. If you are putting it in your posts, you probably don't mean to convey either of these meanings, which is why you're seeing people change it.
an+other is not the same as another.
If you say:
I will buy the other book.
the noun book is definite (it is a certain book even if you don't say so). Therefore the indefinite article usage is incorrect. You have to use a definite article.
But if you meant it to be indefinite (not a specific book), then use another:
I will buy another book.
Definitely it is context dependent.
As the other answers have said "another" is considered the correct spelling, and has been for some time. But as for why?
English spelling has been in flux for a very long time. People, even educated people, did not concern themselves with "correct" spelling until printing became a big deal in the late 15th century. After all, educated people wrote in Latin or French, and considered English to be "vulgar". Around the 17th century, and with growing frequency until the early 20th century, there was a tradition of people who had some standing or felt they had some standing that would write books on how writing "should be done". Geoffrey Pullum nicely sums up my opinion of most of these in his article on Strunk and White's The Elements of Style. "Another" was one of the rules that came out of this time. There is no logical reason for it, except that someone decided that's the way it should be, and their rules were sufficiently respected and passed on that it became the proper way to spell it. "Cannot" is a similar word that many modern grammarians (and spell checkers) try to force on us that I simply can not stand.
It should be noted that there are situations in which separating "an" and "other" is appropriate. The example that comes to mind is "An other worldly sound", where "other" modifies "worldly". This could be emphasized by hyphenating "other-worldly".
I want another X can mean you want one more X. You'll have 2 X's in the end.
I want an other X means you want a different X. You end up with 1 X in the end, throwing out the original X you had. (You'll probably never see or hear this example in real life because it can be very confusing.)
Another X can mean the same as an other X, but not the other way around.