In Auto, they should be similiar...but if you want, the SX260 has full PASM functionally so you have full control of the exposure, and if you really want, you can hack it into shooting RAW.
And from what I've seen, the Sony HX20/30 has better video, and more responsive faster auto focus. But yes, the SX260 has greater latitude for manual controls, and perhaps slightly better image quality.
In that price range, there will of course be some pretty significant pros and cons of each camera. From the size and feel in the hands, to extras like wifi and gps, to the degree of manual control, etc.
In the hands of a good photographer, and in good light, of course you can get good photos with those cameras. Just as you can get good photos with a good smartphone camera.
But I find that those basic types of cameras are trying to compete with smart phones. They go for "good enough" image quality, then try to add in smart phone type features, such as in-camera stitching (HDR, Pano), wifi, gps.... And then they try to "top the smart phone" by putting in an impressive zoom.
But ultimately.... in terms of image quality, in terms of challenging lighting, or fast moving action... the overall performance of those cameras will be pretty similar.
As you mentioned before, the Canon S110 is a bit of the baseline of "next step up" compact cameras.
Another camera I might also consider if I wanted a step up on a tight budget, would be the Nikon P310. Still has the tiny sensor, but it has a 1.8 lens and manual controls. Smaller sensor than the S110, but faster lens and it's cheap.
Sensor size -- not camera size -- is the #1 driving point in image quality AND camera price.
Unless you are looking in the used market, bigger sensors cost more than smaller sensors.
That's why budget compacts try to out-shine smart phones with a big telephoto zoom, instead of a bigger sensor --- Because it's very cheap to stick in a big telephoto zoom, but it's extremely expensive to increase the sensor size.
For anyone looking to put performance and image quality at the forefront, but on a tight budget, my recommendation is always going to be look at the used market/discontinued/discounted larger sensor cameras.