As others have said, money isn't everything.
But if the new employer is offering more benefits, money etc.
It's definately worth looking into especially how you say that your current employer wouldn't think twice about dumping you if they thought they could get away with out you.
4 years ago I decided I didn't like working for my current employer and wanted better. Took about 6 weeks to get offered one of them high paying, comes with a pension gubermint jobs.
I have basically trippled my salary in the last 5 years and am very happy with my employer, so even if I am offered 10 or 20K a year more I will be staying put.
Banks are excellent businesses to work for, especially the big ones as they offer great benefits and a stable work enviroment. When I first got into IT in the early 2000's I did a lot of contract work for BMO and they were very good people to work with.
It sounds to me that you are afraid of the uncertainty of the change, but if you think about it moving to the larger outfit could definately benefit you in the long run as you will most likely be able to get extra training and move up the ladder into a position better than your current one. I was lucky in that it only took me about 6 or 7 years to move up from tech to management.
Good luck with whatever you decide.