If I say that "the resulting object A" and "the resultant object A", do the two statements mean the same thing always, sometimes, never?
Yes the two statements mean the same thing.
Resultant is an adjective formed from the verb 'to result' which means 'that which results' e.g. 'We restructured the business with resultant cost savings'
Resulting is the present participle of that same verb which can be used in a variety of ways, one of which is as an adjective which means exactly the same thing as resultant. Indeed you could simply replace resultant in the above example with resulting.
However other uses of resulting would include:
'He had lived a dissolute life, resulting in serious ill health at an early age'.
'Resulting from the new evidence available to the defence, the prosecution decided to drop the case'.