What is the difference between the two, and which one of the following is correct? 1) Your country's experience of democracy. 2) Your country's experience with democracy.
Both prepositions, of and with work after experience but the choice is likely to depend on the context. For example, it is more natural to say:
I don't have much experience of mountain climbing
indicating that the speaker is relatively new to mountain climbing and implying that s/he would not wish to tackle difficult climbs. On the other hand, you might say:
I don't have much experience with John at the helm
I don't have much experience with this particular model
indicating that your limited experience is not of John, but with his leadership - and with driving a particular vehicle.
Google Books Ngram Viewer indicates that of is about three times as popular as with in this context.
In some examples, including those you give, both prepositions are idiomatic and the meaning is the same.