I had a bank account in the US for a while now, now I am thinking of
switching to a better bank. Would closing the old account affect my
credit score or anything else. I would like to close the old one to
avoid extra paperwork etc. Are there any disadvantages of this apart
from a banking point of view?

Changing banks will not affect your credit score.  
I hear that in the good ol' days it paid to have a long standing
relationship with your bank (they were more likely to trust you with a
good loan)... but now those days seem long gone.  Perhaps if you are
in a small town where you know the managers/owners of a bank
personally then a long relationship is worth persuing, but otherwise
change away.