Nothing. They are also known as "inverted commas".
Either can be used to indicate either a piece of speech/dialogue or a quotation. Essentially, the inverted commas/quotation marks must be the same at start and end of the passage. If used for quotations, within which there is a sub-quotation, then the form not used to open the quote is used to open the sub-quote:
‘HAL said, “Good morning, Dave,” ’ recalled Frank.
“HAL said, ‘Good morning, Dave, ’” recalled Frank.
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