I am speaking of fast road bikes here. Mountain, commuter, and touring bikes with cantilever brakes are a whole ‘nother subject.
I could go on all day about this stuff, I love bikes. I actually shortened the following text for brevity.
Side-pulls starting taking over when Campagnolo came out with the1968 Record side-pull, which was a better brake than than the competition, and finally a side-pull that was relatively easy to adjust and keep centered. The industry followed Campy’s lead.
I think the geometry gives more power with the same hand effort compared to center-pulls, and as @elbanditoroso mentions, the movements are shorter, too.
Side-pulls used to be a bit finicky to keep centered. When the dual-pivot design came out, game over. They are easy to adjust and stay centered. You can see single- vs dual-pivot here at the late great Sheldon Brown’s site.
Center-pulls still have their fans. Center-pulls typically provide more clearance and lots of people are using fatter tires and fenders. Center pulls will mount on a frame with a single bolt, you don’t need the twin mounts as cantilever brakes do.
Here’s a new bike built for center-pulls, the Soma Pescadero. Note the 42mm tires, compared to the 23mm typical on fast road bikes.