The other site is on WordPress.COM – even the same theme is not always identical between the .COM and .ORG versions. So that might be the difference. You might be able to look at all the CSS code for that site and see where they are different.
I’ve changed from the original horizontal top navigation to the left side navigation just to fix the problem for now.
The side navigation pops out the drop down menus when the screen is small/mobile. This will do the job for now but the original issue is still unsolved.