What is the difference between many and plenty?

I would think that "many" refers to a count (large or indefinite number), whereas "plenty" refers to there being "enough" of any given item. For instance, you can have MANY bananas but one banana may be enough.

Many is a relative term dealing with numbers of items. For instance, I can have many apples. Plenty, on the other hand, is more of a survival term. I can have may apples without having plenty. If I have many more people than I do apples, I don't have plenty of apples to go around.

Somebody might have only two cents, which is not many coins, but it is more than plenty to purchase a gumball.

An example of the way language differentiates between things that can be counted and stuff that has to be measured.
The classic case that so many people and businesses get wrong is 'fewer' and 'less'. It irritates me to see a supermarket checkout that is marked 'for less than ten items', when it should be 'fewer'.
I suspect this same difference is the root cause of disputes about when a new century, millennium etc starts. Some people view time as a number of years that can be counted, others as a continuous entity that has to be measured.

There is a counting system that (in translation, of course, runs one...two...many...plenty. Can't find the reference to it at the moment, but see http://www.math.chalmers.se/Math/Grundutb/GU/MAN250/S04/Number_Systems.pdf for a slightly more primitive one.