In my experience, as an American, rich is much more commonly used.
But there is a slight difference as well. The one I choose to use depends on the implied meaning I want to communicate.
"Rich" always implies extravagance but is used for a wide range of wealth.
"Wow, they must be rich."
"Well off" implies the lower range of wealth. One is well off when they are not struggling with debt and have more than they need.
"Don't worry about them, they are well off now."
But those are only the basic uses of the two, without any adjectives or emphasis in tone.
"He is pretty rich." -low to mid level wealth
"She is crazy rich." -high level wealth
"He might not think so but he is well off." -low level wealth
"Listen, she is well off." (spoken with emphasis) -mid to high level wealth
Emphasizing well off like that is often used to speak modestly but mean very rich in certain situations where you don't want to say it directly.
Note: When I say "low level wealth" I still mean wealthy, such as upper-middle-class. But of course it is also subjective, considering someone else as rich or well off depends on your own viewpoint and situation with money.