To me it looks like the US government wanted to double the FBI's size without saying that's what they were doing.
[0] https://www.ice.gov/about/offices/homeland-security-investig...
As for the missions, from both your links:
"The mission of the FBI is to protect and defend the United States against terrorist and foreign intelligence threats, to uphold and enforce the criminal laws of the United States, and to provide leadership and criminal justice services to federal, state, municipal, and international agencies and partners. It performs these responsibilities in a way that is responsive to the needs of the public and faithful to the Constitution of the United States."
"HSI conducts criminal investigations against terrorist and other criminal organizations who threaten national security. HSI combats worldwide criminal enterprises who seek to exploit America's legitimate trade, travel and financial systems and enforces America's customs and immigration laws at and beyond our nation's borders."
Yes there's quite a bit of overlap there. Maybe the key difference is that ICE's mission talks about customs and immigration laws specifically.
What do you mean? The Homeland Security Act of 2002 did pass congress.