In what scenarios would you use 「」 quotation marks instead of 『』, and vice versa?
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In Japanese, the symbols 「」 are called 鉤括弧 (かぎかっこ) and the symbols 『』 are called 二重鉤括弧 (にじゅうかぎかっこ). The basic rules for these symbols are simple: 「」 is used to denote quotation, and 『』 is used to denote quotation inside a 「」-quote.
In some contexts, 『』 is also used to denote the title of a book and other kinds of works.
Example (from Wikipedia with an English translation by me):
I may be wrong about that, but I think they just serve as primary and secondary quotation marks - the same way single quotes and double quotes serve in English. They exist since sometimes printers want to distinguish two kinds of quotes. How exactly they're used and what exactly they distinguish depends on the publisher or writer, of course.
The most usual usage pattern (at least for printed publications) is probably the same as in English: 「 」 are used for main quotations and 『 』 are used for quotations-within-quotations. That doesn't say this is the only usage pattern though.
Since most published writing today is actually out there on the net, and doesn't go through a punctilious (so to speak) publisher, you get all kinds of different and quite unorthodox uses of punctuation. For that matter, respectable printers never used straight quotes ("like these") in English, but only smart quotes (“like this”). They also used em- and en- dashes, never minus signs. But that doesn't prevent people from using straight quotes and minus instead of dash everywhere - so expect the same to happen with Japanese. Standards for punctuation marks existed in the Age of Print, but now the Age of Print is dead, long live the Age of the Internet.