What's the difference between "hundreds of thousands of" and "hundreds and thousands of"? Are they both correct?
The difference is that "hundreds of thousands of" means "at least 200,000", but probably more. It's vague but huge (relatively), but "hundreds and thousands of" is illogical and semantically untenable. If you say "Hundreds and thousands of protesters gathered in the square", you're obviously equivocating ("attempting to deceive").
You can say that "Protesters came by the hundreds and thousands" to indicate, perhaps, the sizes of different groups of protesters arriving at different times and from different places, but it doesn't tell the reader/listener anything about the total number of protesters in the square.