In japanese writing, the ubiquitous 「」is used to mark speech. I have seen "" being used instead of 「」especially on messaging apps. Question is, what's the difference? For the record, I have only seen "" being used in messaging apps
The primary function of quotes is to enclose what someone said or thought, usually along with a quotative-と. For this basic purpose, the "proper" Japanese quotation marks (
「」) are normally chosen, but other symbols are sometimes used in informal situations.
To highlight important or unfamiliar words, various symbols including
【】, 傍点 and so on are used, according to the writer's preference or the rule of a publisher. (Japanese does not have italics.) Perhaps
「」 is the most traditional, but
“ ” is equally common. In general, many Japanese people do not make a strict distinction in the use of these symbols (see this and this) to highlight words, so you don't have to worry too much.