AdEx is focused on improving the digital advertising landscape by introducing a drop-in solution that works similarly to existing ones (e.g. Google Ads), but with privacy preservation and minimized fees.
The Basic Attention Token (BAT) takes a different approach: users of the Brave browser (which blocks all ads) get rewarded for seeing whitelisted ads. Once this happens, they can choose to reward the publishers, and/or withdraw the token. As of 2019, BAT only works within the Brave browser and withdraws are not live yet.
Another prominent difference is that AdEx uses DAI for payments, which is price-stable (pegged to 1 USD).
Those are two completely different approaches, and we believe both can coexist.
It's worth noting that AdEx provides the technical ability to pay users for their attention, but it's up to the publishers whether they will do it, and how.