I've seen both the western comma "," and the eastern comma "、“ used in Chinese before, although the "eastern comma" seems to be used more frequently (correct me if I'm wrong). Which is more proper to use in general?
In proper use of the Chinese language, you're talking about the difference between the true comma 逗號 「，」 and the enumeration comma 頓號 「、」.
The difference between the two can be illustrated in a sentence from the Chinese Wikipedia article on 頓號:
(Enumeration commas are) used when separating similar items consecutively, such as single characters, words, or short sentences/phrases, where the pause is shorter than that given by the True Comma.
The list begins with the word 單字, continues on with 詞語, then finishes with 短句, after which the true comma is used to create a pause between the previous part of the sentence and the next part.
Note: Very rarely will you see enumeration commas everywhere and true commas not used at all. In this case, you're most likely looking at a Japanese rendition of a Classical Chinese text, because in the Japanese language they use 「、」 for everything.
Always use full-width comma ("Eastern") when the context is all Chinese.
However, use half-width comma ("Western") in the following situations:
1. Academic use, when commas appear in formulae and mathematical expressions.
2. Quotations, when you need to quote someone in its original language, which always use half-width comma.
It is worth noting that there is still NO standard usage on foreign punctuations in Chinese context. I mean, it hasn't been standardized on national level.
EDIT: The "comma" in your OP is actually a caesura mark (顿号), which is totally different from the real "Eastern" comma "，".