1. To break a rule
2. To reach across the table for food
are both a reference to a rule in Chinese chess stating that 象 (elephant) is a defensive piece and cannot cross the river, with the latter being more of a metaphor to describe someone who reaches across the table for food, which is considered a bad table manner.
On another note, mainland China (including Canton) actually uses simplified Chinese writing (which came out from the Deng Xiao Ping era) whereas Hong Kong specifically maintians the traditional writing. The difference is literally less strokes and simplification of the characters. The infographic here uses the traditional writing. Personally I feel the two writings are mostly close enough that people should be able to guess what the other means. And AFAIK, it's a 1-to-1 mapping anyway.
Simplified + Mandarin -> Mainland China
Traditional + Cantonese -> Hong Kong, Macau
Traditional + Mandarin -> Taiwan
Also, many of the "simplifications" are based on Mandarin pronunciations, and were quite incomprehensible to me as a Cantonese speaker when I set foot into mainland China for the first time years ago.
How you might write the conversation
"Does he know how to speak Mandarin?
"No, he doesn't."
in Modern Standard Chinese characters contrasts with how you would write
"Does he know how to speak Cantonese?
in the Chinese characters used to write Cantonese. As will readily appear even to readers who don't know Chinese characters, many more words than "Mandarin" and "Cantonese" differ between those sentences in Chinese characters.
Chinese characters still represent SPEECH (not ideas or abstract concepts) and do so in a way that is specific to the particular Chinese (Sinitic) language that one speaks. The long story about this can be found in the books The Chinese Language: Fact and Fantasy or Visible Speech: The Diverse Oneness of Writing Systems by the late John DeFrancis or the book Ideogram: Chinese Characters and the Myth of Disembodied Meaning by J. Marshall Unger. The book Reading in the Brain: The Science and Evolution of a Human Invention by Stanislas Dehaene is a very good book about reading in general, and has a good cross-cultural perspective.
Many other examples of words, phrases, and whole sentences that are essentially unreadable to persons who have learned only Modern Standard Chinese can be found in texts produced in Chinese characters by speakers of other Sinitic languages ("Chinese dialects"). Similar considerations apply to Japanese, which is not even a language cognate with Chinese, and also links Chinese characters to particular speech morphemes (whether etymologically Japanese or Sino-Japanese) rather than with abstract concepts.
The written system in Hong Kong uses the standard "Mandarin" saying. Most serious Hong Kong newspapers (business, etc) "speak" the standard "Mandarin" way.
It's like us learning regular English in school but everyone
speaks Shakespeare or something (of course it's not that
dramatic of a difference).
The Chinese writing system tends to obscure these differences because cognates among the different varieties of Chinese are represented using the same characters, even though their pronunciations have diverged considerably. There are vocabulary and grammatical differences as well, though they are not as big.
The difference is more akin to if English speakers had to write in German†, but because they generally don't know how to actually speak German, they would read it using English pronunciation instead. For instance, if you wanted to say "I know that you said that", you might have to write:
Ich weiß dass du das gesagt hast.
I wit that thou that y-said hast.
† The difference between English and German actually spans more than 1000 years, but I'm not as familiar with other closer Germanic languages with which to provide a good example.
While I think (second hand - I have no deep familiarity with either language) that your broader point holds, this particular bit is weaker than it might be read. While the split happened longer ago, there are more day-to-day attempts to communicate between native speakers of Cantonese and Mandarin than between native speakers of Modern and Shakespearean English. I could see this having an effect on the ongoing development of each language in a way that might leave them closer together than they would otherwise be. (Shakespearean English probably does have some continuing impact on Modern English, but I think we can be fairly confident that there's no influence the other way).
I could see this having an effect on the ongoing
development of each language in a way that might leave
them closer together than they would otherwise be.
For instance, if I take my previous example further, if such a situation existed with English and German, the words "wit" and "thou" might have ended up being more common in spoken English, or at least considered more formal rather than archaic. Certain grammatical constructions might also end up being more common, such as the continued use of the y- prefix in past participles.
I'm already stretching this analogy more that I should, but hopefully it gives a better idea of the relationship between Cantonese and Mandarin.
Many Mandarin speakers see Mandarin as being equal to "the Chinese language" and often think of Cantonese as being "merely a dialect" of Mandarin because Cantonese speakers appear to read and write the way they do (even though the aforementioned examples show that this is really not the case), and they are not really exposed to the more major differences that would exist if Cantonese speakers actually wrote the way they spoke.
Cantonese speakers are more aware that they don't write the way they speak, but because the Chinese writing system only reflects the lexical and grammatical differences rather than the pronunciation differences (even though most of the differences are in the latter), many Cantonese speakers end up considering the written language as being a "formal" version of their language rather than a different language altogether. For this reason, many Cantonese speakers also consider Cantonese to be a dialect as well; however, they consider both Cantonese and Mandarin to be dialects of "Chinese" rather than Cantonese being a dialect of Mandarin.
Linguistically speaking, because spoken Cantonese and Mandarin are generally completely unintelligible to each other, they should be considered separate languages, but the the writing system ends up obscuring these differences. There are also political implications if Cantonese were considered a separate language because of Chinese nationalism tends to see that as a slippery slope towards separatism.
Aside from that this is quite good. I feel like some of these proverbs could be quoted here on HN comment threads.