Is there any difference in meaning between these two above. I only suppose that the first one is more formal.
Yes there is a difference of meaning. The first is the most descriptive.
Approaching the bank means introducing yourself to them and stating your wishes, i.e. asking for a loan. This is a different meaning from "approach" meaning to get closer. In fact you could have approached the bank by letter.
"I liked the girl a lot, so one day I got up enough courage to approach her".
This doesn't just mean "get near to her. It means you went up to her and said "Hello" or whatever.