Having worked at both places in my youth, I can perhaps provide some insight.
McDonalds uses a teflon coated clamshell grill to press and cook the patties. The teflon doesn't let grease stick, and pretty much every patty cooked is like it's being cooked on a new grill surface. I don't recall what the temperature was, but it could cook a grill full of patties in something like 38 seconds.
Whataburger uses an old fashioned steel grill with a burger press. They are cooked the old fashioned way: Put on the grill, cook a bit, flip over, put the meat press on it, finish cooking. It took considerably longer than 38 seconds
At night, this is scraped down with a scraper, but using the day's burger grease to help lift away solids. This allows the grill to get seasoned, much like a cast iron skillet.
I imagine much of the difference in flavor comes from this difference. I'm also sure that the meat formulation (percentage fat, etc) is different, since Whataburger patties always looked a little fattier than McDonalds to my eye. Lastly, Whataburger quarter pound patties had these kind of 'cube steak' kind of regular indentions pressed into the meat, which affects how the things cook.