Why do interest rates vary from bank to bank?

That's one of the ways they compete between each other.

That's one of the ways they compete between each other.

Yes, this is true. Additionally, there is the issue of what money they have available to loan. When that amount is high, they charge lower rates. When that amount is low, they charge higher rates.

The other issue is about credit rating and cash reserves of the borrower.

There is a big range of what banks will charge based on the "whole picture" of the borrowers situation, which is: Why the money is needed, the credit scores, the debt to income ratio, the cash on hand, etc.

So to make a long story short, when you're looking for a loan, shop around just like you would for any other thing that you're going to commit to for big bucks long term.