do (a+b)* and (a|B)* produce the same DFA and same output? In mathematics, wherever the word 'or' is involved, we use addition operator. So does that mean that both expressions are equivalent?
It depends on the context where you get the 2 regular expressions from.
If you interpret both regex in the syntax of real life regex engines, they have different meanings, as Ed Cottrell explained in his answer. + means repeating once or more. | means alternation.
However, they can mean the exact same thing, if you interpret + in (a+b)* as alternation, following the notation in most books on automata theory, and | in (a|b)* as alternation, following the notation in most real life regex engines.