Can anyone explain the difference between the logarithm of a product and the product of logarithms?

Well, for one thing, they are just different.

The logarithm of a product is sort of much more simple than the product of logarithms. Why? Because if you have the logarithm of a product, then the following formula is true:

ln (a*b) = (ln a) + (ln b)

And if you have the product of logarithms: (ln a)*(ln b), you can't do anything with it

The product of logarithm doesn't carry much meaning, it is just some expression. And the logarithm of a product makes much more sense. For example, in Physics there are lots of expressions like ln (something*something), but almost never (ln smth)*(ln smth).

Well, that's not much, I've tried my best. The short answer is: they're just different, and it's easy to find an example when they are different. What I've tried is to give a somewhat bigger answer. As for me (personally, and I do have some experience), logarithm of a product is something "nice" and desirable, while product of logarithms is something "uncomfortable" )