Written and spoken not matching is very common in China.
Obviously, the old-time Danes travelled much farther than they normally get credit for. 
Writing a sign language (for itself to itself, rather than transcribing/translating into another local language) would require a new kind of abstraction - something that would perform the function that an alphabet performs in English or that a character system performs in Chinese - to make it feasible. (Of course concrete methods such as animations are possible, but that’s not writing.)
It would be difficult and disorganized for early adopters, which might be the reason it isn’t already widely used. (I say that only because I have a feeling that someone somewhere has probably come up with at least a preliminary solution.)
- The character meanings are relatively constant especially among writing in different dialects now. The differences in the way you write the same thing is more akin to regional variation in speech in English than to a different written dialects, let alone different languages. And writing has tended to homogenize as Mandarin has became dominant.
Classical or Literary Chinese, which isn’t spoken per se by anybody, uses the characters quite differently than any modern dialect, but they still mean mainly the same thing(s) one by one.
- Likewise, known character meanings doesn’t get you overnight to reading Japanese or Chinese if you know the other. But it’s a huge head start.
I read Japanese passably for my purposes* based on knowing Chinese character meanings from Korean (which I speak OK, not a native speaker). I only speak a little Japanese, much less than I can read, because I usually only know the meaning and Korean sound of a given character not the Japanese pronunciation (which also varies by case, usually doesn’t in Korean). I’ve also fumbled through Sun Tzu in the original Literary Chinese. Again nobody speaks LC, but I don’t speak any Chinese dialect. I also read modern Chinese writing to a limited degree (trying to improve). Character meanings get you most of the way. Then you do have to pick up on kana in Japanese case, and difference in character forms from traditional form used in Korean (not commonly used nowadays, but which still underlie most of the words in the language). But starting from scratch is an all together higher hurdle.
The character forms are like fonts, the way computers now understand them. Simplified and modern Japanese are different ways of writing the traditional characters, not different characters. Characters unique to Japanese are few.
*study of the Pacific War or other Japanese military topics in Japanese language sources.
Not really different systems, all of them can be (and two almost always are) combined into one sentence.
Kanji - characters adapted and adopted from written Chinese back in the first millennium (and later). The meaning often relates closely to the original Chinese, but not always. The keystrokes may vary from modern simplified Chinese characters.
Kana - two types, both different ways of writing the same syllabary (like an alphabet, only each one stands for a syllable instead of a letter).
Hiragana is used for native Japanese words; they can replace the sound values of Kanji but that may leaving the meaning ambiguous; also very widely used for necessary grammatical things like declensions and sentence particles. Hiragana can generally be read by the poorly educated who don’t know very many kanji.
Katakana is usually used to transliterate (usually poorly) the sounds of foreign words, limited because there are fewer vowel sounds and consonant-vowel combinations in Japanese than some other languages. May also be used for emphasis of a Japanese word, sort of similar to italicizing an English word.
Roman letters (romaji) are also used, either to print foreign words directly, or to transliterate Japanese words into Roman letters (most common example are railroad location signs that are printed in Kanji, in hiragana (for those who don’t know Kanji) and in romaji (thank you General MacArthur)). This isn’t really Japanese at all, but Japanese people do use them and most can read them enough to put the sounds together.
There, aren’t you glad you asked? You didn’t ask? Oh, well then, never mind.
My anecdotal evidence on this is that a couple of non-Japanese speaking Chinese students were at my Japanese language school in Japan, and although they learned the spoken language about the same as everyone else, they had a big head start in reading and writing. Even if some characters were not identical or there had been a meaning shift, at least they were used to the forms and could learn them more quickly.
And people who have had a few semesters of Japanese. I need to re-practice.
Katakana is also for onomatopoeia, isn’t it? And a few weird cases, like “Mazda” is both from a name (Matsuda, would be 松田 in Kanji) and a god (Ahura Mazda).
And romaji itself has several systems, broadly two categories: Western-influenced forms that preserve the pronunciation (Hepburn and variants) and native forms that keep the romanization uniform. I prefer the former, as the hypothetical romanization “paatii” could be pronounced the same and written パーティー (party), or pronounced “paachii.”
Japanese is a pretty easy language to start learning, very difficult to master. Which system to use is pretty easy to pick up, even if learning Kanji is hard, not to mention all the grammatical and social nuance. Chinese is not quite the opposite, but people seem to struggle more at the beginning, and advance faster once they get tonality etc. down.
Does everybody in Norway know how to read and understand Nynorsk and Bokmal?
Norwegian dopers are probably getting ready for bed, but AIUI it varies but most schools use the “official” local form in most cases of instruction. Here’s a map; Nynorsk is blue and despite the large area, is the lower population part, so under 15% of people use it. The Nynorsk area covers Bergen, the second largest city, Oslo is in the red-orange/gray area in the inlet (“Oslofjord,” though it’s not nearly as cool as real fjords) to the southeast.
Yes, but there is still grammar. If you translate word for word from one language to another, it often comes out horribly.
French: Je ne parle pas Francais
English: I no speak not French
English: I do not speak French
French: Je fais pas parler Francais
Wouldn’t Mandarin and (e.g.) Cantonese writers just seem barely literate to each other?
So other language speakers learn to use Mandarin grammar when they read and write? That’s interesting, thanks.
The simple answer is that time, distance, and isolation cause languages to shift. At one time (IIRC, commented on by Samuel Pepys) the different areas of England had wildly different languages. It probably didn’t help spread a common language, that for a long time, the hoity-toity types spoke Latin and (Norman) French. But think how different Appalachian or southern US people sound, even though they’re speaking the same language. Then consider the Australian accent, which almost sounds like a different language (a common vein of humor…). Even more so, Newfoundlanders can be unintelligible. Things drifted over time, and nobody goes around teaching the “proper” way.
But then, that’s how we ended up with Spanish, French, Italian, and Portuguese instead of Latin; and apparently written Latin based on Republic speech was wildly different by the third century AD from what the plebs spoke.
Nowadays, the trend is opposite. Everyone hears the same bland mid-American English in the media unless it’s specifically poking fun at an ethnic pronunciation, thus encouraging and teaching people to speak closer to “the norm”. Similarly, even a lot of minority languages are dying since the locals end up fluent in the major tongue too and eventually find fewer and fewer people to use the original language with.
(As a note, when I was on a trip in Italy many years ago - many of the pop songs on the car radio were in English. What does that do for a national culture when the younger generation experiences more and more in an international language, rather than their native one?)
In fact, one book review I read in Economist suggested that you can find a number of locales in China where the grammar and idiosyncrasies of the local dialect suggest that at one time it had been a completely different language, but the words imposed from above had replaced the language, just not the way of speaking.
IIRC, the opening words in “Mon Oncle Antoine” were:
“Hey Joe, le boss…”
In fact, a major impetus in Quebec politics is to enforce the use of French as much as possible to prevent it from being swept away in a continent where the 360 million people surrounding them overwhelmingly speak English, and the media bombards them with English ll the time.
A few people have come up with ‘phonetic’ systems for writing sign languages (if you think of sign phonemes as consisting of handshape, orientation, movement, location and facial expression) but they haven’t really caught on as yet (if ever).
Signwriting
ASLwrite
That’s not (strictly speaking) writing at all. The essential thing about writing, the thing that makes it be writing, is that it’s abstract and arbitrary. Drawing pictures of how your mouth looks when you speak is not how English writing developed, and drawing pictures of how your hands look when you speak is not going to be how ASL writing develops either.
Right, now there is basically only one acceptable form of written Chinese, which is ‘Mandarin’ (Putongwa, or common language), even where the pattern of spoken dialect, not just the character pronunciations, would vary.
But this isn’t really new, extending much further back and also outside China under the previous written standard of Classical or Literary Chinese. That language also didn’t correspond to how many or eventually basically any people actually spoke but was standard for writing. And also beyond China. Literary Chinese was used in ‘serious’ writing (like official court documents) in Korea till the late 19th century, several centuries after the invention of the hangul alphabet. It was likewise extensively used in Japan and Vietnam. People in those countries if they read LC aloud would use their own pronunciations of the characters, like different dialect speakers in China. It’s role somewhat resembled that of Latin in post-Roman Europe for many centuries.
But LC is much more different than any modern dialect in how it uses the characters (albeit their meanings remain generally similar) than the modern dialects are from one another. I think the official idea that there is now a single Chinese language with different dialects is basically correct.
Personally, I don’t think a writing system for ASL will ever be widely used. There is no ‘market’ for it and high barriers to introduction.
But there is nothing to say that different writing systems all have to develop the same way. How your hands look in ASL does in fact determine what sign you are making, the same as making a ‘b’ sound or a ‘p’ sound determines what word you are saying in English. You could make squiggly, abstract signs for ‘index finger point’, ‘fist shape’, ‘flat hand’, etc. but there is no reason they have to be abstract. There is an argument that making them visually descriptive makes them easier to learn and easier to adapt to different sign languages, which will have different handshape repertoires.
I saw another writing system once that was more focussed on making the ‘letters’ easier to write, but I don’t remember the name of it. It was only in the early stages of development anyway, and still had a long way to go.
In particular, in Japanese you can’t have a syllable which ends in a consonant (except for N, which Japanese treats differently from other consonants). So foreign loanwords often end up with superfluous (to Western ears) vowels stuck on at the end, or inserted between consonants in a blend.
I’m told that this is also sometimes done just to be trendy, in much the same way that Americans will get tattoos in Asian languages that they don’t themselves speak.
When I first started watching Hong Kong movies, I couldn’t understand why the Cantonese soundtrack didn’t match character for character the Traditional Chinese / Hanzi subtitles, while the Mandarin soundtrack did. BTW, I don’t speak or understand either dialect or read Hanzi, but followed noticed the discrepancy because both Chinese and English subs were displayed at the same time.
I later learned that Cantonese contains phrases that can’t be expressed with Hanzi or directly translated to Mandarin. Here’s an interesting and funny video comparing Cantonese and Mandarin with some examples:
https://www.youtube.com/watch?v=e73btaVo868
Interestingly (at least me 
) is the theory that some classic Chinese texts are written in a style based on a variation of Cantonese, particularly poems that don’t flow when recited in Mandarin.
As for the original question, I suspect major reasons are secrecy, fellowship (group and national) and non-local communication.
Secrecy - Sometimes children (and even adults) will develop their own “secret” language so others won’t know what they’re talking about. This is advantageous in keeping your neighboring country (who may not always be neighborly) unaware of your strengths and weakness.
Fellowship - Tied into secrecy, developing your own language or dialect makes it easy to identify your fellow members, whether in a group or in a country. I’ve read that some languages (and I suspect the majority of them) have at least one word or phrase that describes those outside their realm collectively as “them” as opposed to “us”.
Non-local communication - This is particularly clear in Hawaii, where Pidgin English evolved as a way for multiple immigrants to communicate. Even now, Pidgin English is still evolving with different phrases (dialects?) according to location, even within the same island. Even as a local born and raised in Hawaii, I have difficulty understanding some of the phrases used by “heavy” pidgin speakers or those from neighboring islands.