If you only could visit one which would you pick and why?
How important are sunny beaches to you? Madeira is drop-dead gorgeous, but it has no natural sand beaches. Not being a beach person, I hope to make a third trip to Madeira at some point. I haven't been to the Azores.
There are few actual "beaches" (as most of us would define them) in the Azores. Most of the coastline is volcanic rock, always hard, often sharp and unforgiving. There are a handful of real beaches on Sao Miguel, and people do swim from them, but the "sand" at them is dark, coarse, ground-up volcanic material. Swimming in the ocean is definitely popular, but typically done from what locals call "natural swimming pools", rocky tidal pools along the coasts with ladders and occasional diving boards (as one sees in any large swimming pool) installed in the black, volcanic landscape. They're fun and refreshing to swim in (people go there and hang out, set out towels, lay in the sun, etc., just like at a conventional beach or swimming pool), but it's a bit odd - they're quire popular (occasionally crowded) and it doesn't have the feel of a beach experience (at least not one that most of us are familiar with).
The Azores are a poor choice if you're looking for a "beach" experience. Haven't been to Madeira (yet) but I suspect the "beaches" there are somewhat similar.
Beaches aren't my priority. Scenic landscape and hiking are things I enjoy