The difference is a very simple grammatical one. "Stuff" is an UNCOUNTABLE noun. It is grammatically singular. We often use this word to refer to uncountable substances such as liquid e.g. "What's that stuff in the bottle?". "Thing" is a COUNTABLE noun and it is used to refer to other countable nouns. To ask about one individual object ( e.g. a toy) you would say "What's that thing?", and to refer to several individual objects, you would say "What are those things?". With more abstract concepts, you can choose one or the other, but the grammar is different - for example, "There is stuff we need to talk about", "There is a thing we need to talk about" or "There are things we need to talk about." The grammar of the sentence has to agree with the choice of noun.

Stuff is multiple things, but you use it with singular grammar. Like "that stuff", or "This stuff is in the way". Thing is one ("That thing", "A thing I'd like"), and if you say things to say multiple you use plural grammar. "These things", "Those things look nice". Also thing has a special meaning in certain expressions, if you say "He plays baseball in the rain, that's his thing", it means that him playing baseball is something that is unique to him or that he specializes in (like he is very good at playing baseball in the rain). Hope that clears it up.

They are similar and can be used interchangeably I believe. I can't think of an example that specifically demands that you use one or the other.