You asked a very interesting question :) However, as in English, Chinese classical poetry (Tang Dynasty) are not written in the vernacular form but rather in literary Chinese and so there is no great difference between the individual sounds or meanings in themselves. However, the slight change in the combination tones that makes it mutually unintelligible in the oral form. Here is a link to illustrate this point. https://www.youtube.com/watch?v=mmUZH5Zb2kI There are certain words whose meanings are no longer valid in the current form. For example, 走 in Mandarin means to walk but "run" in literary Chinese. This classical meaning is still preserved in Cantonese vernacular usage and in some Mandarin terms. For example, 走狗 = literary running dogs = lackey, traitor. So in the classical line, 走馬蘭台類似蓬 it means "a running horse" to denote speed and hurry. Because of the rules of rhyming and matching/contrasting tones, some characters has an additional sound that is not used in daily life. For example, 黑 (black, "hei") is pronounced as "herk" in poetry, a 去音 no longer used in modern Mandarin. The Cantonese equivalent, "hark" is preserved even in ordinary use - hence Cantonese sounds better than Mandarin when reciting classical poetry. A student learning classical poetry will have to learn all these rules. It is just like we have to memorize that "quay" is pronounced as "key" and not otherwise. Some terms have a different nuance or meaning used in modern usage. For 可憐 ("to pity" or "to be pitied"). It means "can be loved", "to be loved". It is like "vulgar". The older meaning is "common" -- still retained in terms like, "the vulgar world" or the "vulgarity of the people". If you specifically write a poem or song in Cantonese, then those who know Cantonese will know what it means since slang or daily vernacular use is employed.

When learning traditional poems and articles, yes, students are taught the meaning and usage of the words at the time when that piece was written. Jeff has given an excellent explanation:)

Indeed, it is like reading Shakespeare. But it is not always true that Cantonese sounds closer to "early Chinese" or bettern in reading Tang poems. The term 'early Chinese' is hard to define. Mandarin is not a completely manmade accent but based on the regional accent which was once part of the kind of official accent in Ming dynasty as well. So in some poems, you might find Cantonese sounds better, but in some others Mandarin might be closer to its original rhyme. There are many existing researches on this topic but I guess most of them are written in Chinese:( As for the different words, yes, Cantonese is probably the only dialect in China which has preserved the writting form of its own words.