The term "system test" is defined in IEEEStd829™-2008 as
Testing conducted on a complete, integrated system to evaluate the system’s compliance with its specified requirements. (adopted from IEEE Std 610.12-1990 [B3])
whilst systems integration testing is defined as
Testing conducted on multiple complete, integrated systems to evaluate their ability to communicate successfully with each other and to meet the overall integrated systems’ specified requirements
So "integration testing" (regardless if do it in a "Big Bang" manner, or not) means only to check if different modules work together as intended. That does not include to fulfill the business requirements. The latter is what "system testing" is for. (See also Wikipedia).
Let's say you write a chess program. "Integration testing" could be done by some tester who does not know the chess rules completely, only checks if the UI works together with the backend in a sensible manner, and if the system cannot be crashed through the UI. "System testing" then may include to test all the gory details of testing the chess rules, or if adjusting the strength level of the chess AI through the UI works as intended.