Full frame sensors yield sharper images in standardized tests like the ISO 12233 crops and Imatest SFR tests. The reason is the bigger sensor - with the framing matched, the APS-C image must be enlarged more for a given output size, meaning a softer result. The increased sharpness with FF is evident in real-world shooting, too.

In focal length-limited situations, where you could fill the APS-C frame but need to crop the FF image (can't get closer or use a longer lens), the APS-C sensor would *theoretically* deliver higher resolution. However, ISO noise plays a role in perceived sharpness. I've found that in practice when comparing current/recent sensors, an image from a FF sensor cropped to match the APS-C FoV has equivalent IQ at up to around ISO 800, and the cropped FF image has progressively better IQ as the ISO goes up from there.