Every country has a regulatory body or an association that issues and maintains a registry of these numbers.
In the United States the numbers were originally issued by ABA [American Banker's Association] to identify a Bank so that paper checks could be routed easily. These numbers are also used by Fedwire and ACH. The article on wikipedia explains this in more detail.
There is a code in the Card numbers issued by Visa and Mastercard. This was again to help identify the network where the transactions need to be settled and avoid duplicate numbering across Visa and Mastercard.
As the Routing numbers on Banks are used predominantly in the US for US payments, there is not much of an advantage. However it can be of help to figure of some of the scams in place. For example if one receives an message saying you have won something and requests a payment giving the Bank details of a leading Bank [say Bank of America], however the routing number is of some small time credit union; then its a dead give away.
The similar example for SWIFT BIC ID's used for international payments, where one is requested to make payment to a Bank in US or UK, the unsuspecting user who believes that there are enough regulation in these countries may make a payment, however the BIC given in the instruction would be of a Bank in some African country. So knowing the numbers definately help in certain cases.