English: measure certain mass nouns or classify certain nouns. Not used for counting every noun.
Korean: Always used, regardless if they measure or clarify. Six dogs, three months, two papers, four volumes, one bowl, ten things all use counters. Always.
The fact that they are used for all nouns [1] in Chinese / Korean and only one of the two classes of nouns in English is not a difference in the way they are used or the function they perform. It is a difference in which nouns require their use. They have identical grammar and meaning. There is no reason to consider them distinct phenomena; they are one and the same.
[1] Certain things that you might think of as a noun based on English are, in Chinese, measure words which do not measure a noun. For example, 天 "day" and 次 "time" (as in "it happened three times") are syntactically measure words, but it is not possible to follow them with a noun that they notionally measure. This isn't really distinguishable from saying that all English count nouns are really measure words.
English: measure certain mass nouns or classify certain nouns. Not used for counting every noun.
Korean: Always used, regardless if they measure or clarify. Six dogs, three months, two papers, four volumes, one bowl, ten things all use counters. Always.
The difference is night and day.