In this controversy, Jefferson interpreted the Constitution very strictly while Hamilton interpreted it loosely. Jefferson argued that anything that the Constitution did not explicitly allow could not be done. Hamilton argued that anything that the Constitution did not explicitly ban could be done.
This was the first major debate over how to interepret the Constitution. Jefferson wanted to be a strict constructionist. When he looked at the "necessary and proper" clause, he emphasized the word "necessary." He said that the Constitution only meant that things that were absolutely necessary could be done. Hamilton, by contrast, argued that it was obvious that any government had the power to charter a corporation like the bank, whether it was explicitly stated in the Constitution or not. He said that any power that was related to (as opposed to necessary for) an end that was specifically allowed in the Constitution could be done. Therefore, he said, a bank could be created because it was related to things like collecting taxes and regulating the currency.
The major difference, then, was in whether the Constitution should be interpreted strictly, with only explicit powers being allowed (as Jefferson wanted) or whether it should be interpreted loosely, with implied powers being allowed.