Did you turn off your note before swapping batteries?
Were you running a custom rom?
I had the same thing happen to me. I have two batteries (stock and anker) and switch between them everyday. Two weeks ago, I switched them out and it won't turn on for longer than 3 seconds.
From what I have read, it is a faulty power button issue. It could just be you need to get a stock battery. It could be something worse. Update this thread and let us know how it turns out.
---------- Post added at 04:58 AM ---------- Previous post was at 04:15 AM ----------
In the 15 minutes since responding to you, I fixed my phone (it seems).
I took the back off (again!) as per the how-to youtube video....
Then I took some tweezers....
and wiggled the crap out of the power button. Side to side, back and forth....I wiggled with some force. I half hoped I would bust the damn power button off and just use usb to start the thing up....a working note, even crippled by usb start up, is better than none right?
So, needless to say, I fiddled then checked it with a battery...fiddled and checked it with a battery and when I could check it with a battery, where it wouldn't buzz and shut off immediatley, I knew I was getting somewhere.
I am back in CM 10.1 with Flap kernel.
The moral? Grab the tweezers!