It's probably personal preference, but sample images on the Canon S120 look better to me than the Rx100 III.The image quality difference in this comparison has absolutely nothing to do with the fact that the better performing camera is a DSLR. The difference is primarily the result of the sensor size difference and possibly somewhat the difference in the optical quality of the lenses.
You can get a substantially more compact camera which the same "larger" sized sensor in a much more compact camera by choosing either one of the "mirrorless" interchangeable lens cameras from Panasonic, Olympus, Sony, Fuji... or one of the premium fixed lens offering that use the APS-c size sensor (e.g. Fuji Finepix X100S, ...). Even the better cameras with slightly smaller sensors (e.g. Sony RX100) perform excentlly.