Where are these applications (presumably, applications a bank would be interested in) that aren't possible with fiat?

Vault with insurance: Isn't this one of the services banks have provided from the very beginning regardless of whether the currency is gold, silver, fiat dollars or whatever else?

Invest and/or loan: I put these two together because, between the two of them, this is how nearly every commercial bank in the world makes the majority of its profits. You deposit money, they loan it out or invest it, usually they loan out more than you actually deposit (sometimes up to 10x more). Bitcoin is far worse (for the bank) than fiat in this respect, since the bank can not create BTC it doesn't have and loan it out as if it really exists. I guess in theory they could operate a fractional reserve, but with a hard limit on the number of Bitcoins that can exist I see that scheme collapsing pretty quickly, Chase and Bank of America would be in no better position than MtGox because they still can't create these extra BTC out of thin air and unlike dollars, nobody is going to print more once we hit that hard limit. This also makes the "vault with insurance" service much less profitable for the bank than it currently is with fiat money.

I can't think of applications a bank would be interested in that aren't possible with fiat, but for those applications and services that a bank traditionally already provides, BTC would put the bank at a significant disadvantage compared to fiat.