There are lots of audio videos on YT as probably expected but I've just watched something that states there is zero audible effect between a 1/4 inch and a 1/8 inch jack which I agree with totally but as 1/8 inch jack is way more popular for most devices, why do some headphone manufacturers (Sennheiser hd800 for a start) use the bigger 1/4 inch jacks?
I expect it is because the larger jacks are more dimensionally stable - less flex, less wear, etc. They hold up better over time. Which is why most larger amplifiers use them as the preferred output - and most professional gear. So it makes sense for professional headphones to match what their intended audience uses.