The same question can be extended to nearly every language or language family. Why is Basque considered a single language when the most divergent Eastern and Western "dialects" are hardly intelligible with each other? Why was Moldovan often considered a distinct language from Romanian when the differences are far more negligible? Why is African American Vernacular English not a separate language?

The answer lies in the definitions language and dialect. While the two words may imply different things, the sorry truth is that there is no absolute definition of either for the linguist. Thus, there is no line drawn in the sand where a dialect one day may change one inch too far and find itself a new language.

There have been several attempts to provide a distinction between dialect and language. The most popular among regular folk is the 'intelligibility test,' where if two dialects are no longer intelligible between themselves then they have crossed into separate languages. That would be easy, but unrefined. One problem is that intelligibility is often rooted in phonological similarities (not necessarily lexicon- or syntax-based). For example, to a Spanish speaker, Brazilian Portuguese tends to be harder to understand than Portuguese from Portugal, even without significant prior exposure and even if the sentence constructions in Brazilian or Portuguese Portuguese are the same. The simple fact that Brazilian Portuguese has a more divergent phonological inventory is the blame. The same problem can be found in English. Why is Northumbrian English (a dialect) less intelligible than Scots (a separate language) even though Scots has far less vocabulary in common with General American or Received English? And while Northumbrian and Scots are difficult for an unexposed American listener, we can blame phonology for the incongruity.

So the Pop Linguistics intelligibility standard fails as a litmus test, but even more refined definitions have their own problems. Perhaps the most popular criteria among scholars is Bell's Seven Criteria, published back in the 70s. To summarize: language and dialect is contingent upon cultural standardization and identity. Bell's Criteria runs into problems of its own - the foremost being that it cannot wrest itself from the inherent subjectivity of its speakers - but Bell has introduced something novel and important, that political sentiment can define language as much as linguistic differences can.

So to paraphrase John McWhorter, the root of a separate language is as much political as it is linguistic. It cannot be weened from the identity of its speakers, and as a result, there cannot be a purely objective litmus test. So the answer to your question, 'Why is Afrikaans considered a language, rather than a dialect of Dutch' is: Because its speakers consider it so.

Please feel free to pop in to /r/southafrica or /r/afrikaans should you have any more questions on this awesome language.

Adding my two cents worth as a South African who can speak both English and Afrikaans:

There are a lot of pronunciation, vocabulary, and syntax differences between Afrikaans and Dutch which make the languages distinct. While I can more or less understand some Dutch if I read it and pick up a few words of spoken Dutch, I would never say I could speak Dutch.

The differences between English in South Africa and English in England is much smaller. Although our use of English is unique because of our accent and vocabulary (which is influenced by all the other languages people speak here in SA) it is still comprehensible to other English speakers.

Afrikaans and Dutch is really not comparable to the different English dialects, since the languages have grown apart.

May I ask who considers Australian English a separate dialect of English, (rather than just a particular accent of English, as it so obviously seems to be) and why?

Or are such distinctions completely subjective and arbitrary anyway?

I have read many of the excellent answers here for what constitutes Afrikaans as considered to be a separate language, but what kind of things actually make all of the various English dialects all still be considered one and the same language? This is including the Australian dialect along with Jamacan, Dixie (aka U.S. Southern), or the various dialects of the British Isles?

In this case, even the quote that a language is a dialect with its own army and navy doesn't hold water, as there are a multitude of independent sovereign countries each claiming English as a language, where there are definitely people between those countries who literally can't understand one another.