The bank offers several advantages, most notably secure storage of items and money. Bank accounts are per-character, and cannot be accessed by anyone else.
Up to 5 items can be stored in a bank, though not all items can be stored (for example, logs are too large to fit). Containers must be empty when deposited. There is no decay on stored items.
A character must designate a settlement token to use for bank actions. Items can only be deposited and withdrawn at the designated token. (It is not necessary to be a citizen of the settlement.) The bank can be moved to a different token at any time. The transition takes 24 real life hours, and the items inside are inaccessible during the process. If your bank account is at a deed which is disbanded, your bank account can be transferred to a new deed instantly.
The item storage is not shared between servers. This means you can't access items you stored on a different server, but neither do they count towards the 5 item limit.
You can also keep money in the bank. Money can only be deposited at the token designated for item storage (see above), but can be withdrawn at any settlement token. This means you do not have to withdraw money before going on a shopping trip, but you have to make it back safely if you sell something.
Unlike items, deposited money is shared among servers.