Depends on the car, and the situation.
Do I care about either in my old Ramcharger? Not really, but I do like that it surprises people from a stop light. My 8.1 Chevy? Same, but I will take it up to a higher speed on the freeway because the crew cab is a tad more stable over the 80 mark. 8.1 burnouts are infrequent because the rubber costs too damn much but also entertaining.
My 2013 AWD? I don't actually care about 0-60 because where I live, not much anywhere to do it. However, when the opportunity or need arises the AWD means there is no burnouts even with a sudden pedal to the floor. It pulls very hard from 0-115. Then the nanny takes over. In BFE WI, on county trunk roads there are plenty of safe places to push the car, but there is no need and I get little thrill over "top speed". Especially when it is simply the nanny computer metering that speed, and most likely the satelite radio antenna narcing on me to my insurance company. From what I can tell, shorter bursts of pedal to the floor OK, extended periods not OK, based on the results of my premium lol. Not sure what the threshold is but I imagine more than 15 seconds must qualify.
I had a 94 5.0 Mustang. It was stout enough 0-60, did burnouts, and was a proper highway cruiser up to about 80mph with the gears I put in. Very entertaining on the windy backroads. Sold it to my son.
So it depends on the ride. I like acceleration, but not some specific benchmark. Rowing a 5 speed is good fun without any consideration to both.
Burnouts are righteous, and correct, as I have written in other threads. In fact, their is a social obligation to instruct today's youth on how and why burnouts are righteous through proper demonstration and if you have the means you should fulfill this obligation.