They are very similar
You can say:
'That stuff is old'
'Those things are old'
You can't say stuffs, stuff is plural.
You can say thing:
'That thing is old'
I guess things is slightly more polite than stuff.
Stuff is an uncountable noun, so (as Michael has pointed out), "stuffs" is not possible.
Stuff is also colloquial speech, so you wouldn't use it in writing unless you wanted to sound very casual. "Thing(s)" is your standard word, and if you want to be even more formal, use "item(s)".