Note for me ( try to learn en and this work) :\

which tool is better for performing regression and functional testing for banking domain?

"the main question is: is it the database to test or is it an client application to test?"

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1.

if the databases are tested then SQL queries saved to use later are good in regression and functional testing

however if we are talking about some app(ie: brower app to use later are good in regression and what can be automated is good:

Selenium for WebApps, TestComplete or something similar for the rest.

2. 

it's not easy to say but...

as we know - all the banks works on database which have table for accounts, deposits, customers, charges, etc

everything what is made in a bank is written in a database, so..

if we want to see if some charges are taken from account 1234 we can see in the table where have account history with account_nr = 1234 and date = [date of charges]

this is testing uusing SQL queries with database

the same problem with app used by worker of the bank

we must find account 1234 (most cases by the customer), then go into the history of this account and find the history in a day [date of charges] to see is it ok or not.

when we know we will recently check it we can save the SQL query for the later use (database make automatic script (ie. with selenium IDE) which will look for the result automatically.

practice with on discuss

search and research Parasoft SOAtest & Virtualize ( this's a financal organisations for API testing which has been recognised by Wealth & Finance International magazine's Wealth & Money Management Awards 2014 and being named Most Innovative Software Vendor).