I have both for my A6--the Michelins on the factory 20s that came with the car and the Contis on the aftermarket 20s I picked up. I drive spiritedly and tend not to drive in snow (thankfully can work remote in those cases).
To me, the Michelins definitely have less sidewall flex, which means I can corner better and at higher speeds. They do at least as well in rain and better in dry than the Contis. They also have more wheel protection than the Contis do. Noise-wise, they feel about the same? Contis might be a bit quieter on new/fresh pavement. Comfort is about the same, but probably the Contis have a slight edge for me because they are 265/35 vs 255/35, so a bit more meat on them.
Prior to these sets of wheels, I had Michelin Pilot Sport A/S 3+s. They sold me. I had to travel from ~Baltimore to WV and back for work (about 4 hours each way, up/down some good inclines). The tires were near end of life (maybe 3-4/32), and I had ZERO handling in the wet weather at 80-90 mph going up and downhill. Phenomenal tires. Will be missed.
A 3rd and 4th option you can consider would be Vredestein Hypertracs and Bridgestone RE980AS+. I briefly ran the former (they have a 100 day trial period) and the only reason I didn't stick with them was I had noticed my TDI's fuel economy dropped a bit, so I went with the Michelins. Which didn't seem to help there either. But they felt nearly as good as the Michelins in the dry and wet weather and cornering I put them through, as well as being quiet. Never tried the Bridgestones but read good things.
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