In meaning, no not really. Grammatically though "things" is a plural noun and countable, but "stuff" is a non-count noun.

Yes has , you can count things , but cannot count stuff .Stuff is an uncountable noun you can only measure .

'things' is normal but 'stuff' is very casual; not really used in polite situations but very common with friends and family/normal people, ha ha. :)

Yes, "stuff" is said by people who like to boast.

Here is a good explanation: https://dictionary.cambridge.org/amp/british-grammar/uncountable-nouns/thing-and-stuff