The greatest benefit of the bank, as far as I can see, should be the to the guilds.
For instance a guild can have a large bank account that members can draw on to buy stuff, ie: you are in Oja and a player comes in with 68 gold ore to sell but you don't have enough money on you, you pop down to the bank and draw out a temporary sum from the guild account and buy the gold, smelt it, sell it and return the capital you withdrew with an extra guild fee for the use of the account. Of course this does require that all guild member can draw money, to be fixed later I understand, and maybe with a fixed draw limit for each member set by the guild master so that one disgruntled member can't go in and pinch the entire guild capital in one hit, and more branches, again a future feature.
As far as item storage is concerned it seems to me the bank shouldn't be doing that except for maybe gold and maybe silver, you should simply be able to walk into a warehouse or storage area and talk to someone like Jayose and rent a storage locker for a fixed fee.