Quotation Marks / Speech Marks (” “) - As their name suggests, quotation marks indicate direct quotations. You can also use them to show that a word or a phrase is being used ironically, or for titles of articles, book chapters, episodes of a TV-show, etc.
For example:
“You’ll never believe what happened last night,” Jerry said.

Apostrophe (‘) - An apostrophe has two very important uses. Firstly, it can be used in contractions in place of omitted letters. Secondly, it can show possession.
For example:
You don’t have to go to the supermarket if you don’t want to.
We need to invite both of Sally’s sisters to the party.