Not as much as you'd expect. For normal reading and browsing, only stuff behind cloudflare gives any issues (unless you're running a local proxy to handle ssl). "Webapps" obviously don't work, but for everything else it's just a maybe-uglier version of the normal internet. Dillo isn't as good at paywalls as links/elinks for some reason though.
> only stuff behind cloudflare gives any issues (unless you're running a local proxy to handle ssl).
This used to be a problem because Dillo didn't implemented SNI, but should be okay now. If you still have some site that doesn't connect with Dillo from git (there is an updated dillo-git AUR package in Arch Linux), please report it :-)