How to effectively determine if given truth table is equal to another one (when we take into account that they can differ at order of inputs)
They do not represent the same circuit because swapping the inputs may not yield the same result. The inputs are not equal otherwise swapping them would have no meaning.

Consider the case where ABC = 100: in the first circuit it yields 0, in the second it yields 1. So the logic is different even though the gates are the same. If the order of the inputs did not matter, how would you differentiate between ABC = 100 and ABC = 001 since they produce different output in your case?