Communication Complexity for Product Distributions
Set disjointness is easier for product distributions since the hard distribution for set disjointness is very far from being a product distribution. What do we require from a hard distribution $(X,Y)$ for inner product? We want each of $X,Y$ separately to be quite random, and we want $X\cdot Y$ to be mostly zero, say $X \cdot Y$ contains at most a single $1$. This cannot be accomplished by a product distribution. While you can get the $X \cdot Y$ property, this implies that each of the two inputs will be very biased. Conversely, if the inputs $X,Y$ are quite random, then $X \cdot Y$ will have many $1$s.

The inner product function doesn't suffer from the same issue. Indeed, it seems that the hardest distribution is the uniform distribution. You can prove a linear lower bound using the discrepancy method – this is standard, and uses a bound on the discrepancy known as Lindsey's lemma.

Sherstov came up with an optimal gap example in his paper Communication complexity under product and nonproduct distributions. His function is a random function, chosen so that there are no large monochromatic $1$-rectangles. The end result is a function whose randomized communication complexity is $\Omega(n)$, but for any product distribution, the randomized communication complexity is $O(1)$.