There can be significant image quality differences between similar small-sensor cameras. While it's true that brand may not have a significance (a lot of makers use sensor hardware from Sony) beyond processing algorithms, there are two factors that can have a significant difference in the image quality the camera produces, and both of these may not necessarily be easily revealed by the type of camera or looking through the specs.

Lens

The lenses can vary widely both in capability and quality on small-sensored compacts. Some go for superzoom performance, but have limited low-light capability, while others can sacrifice zoom range for fast max. apertures. While two small-sensored compacts may look similar, they may have very different lenses (e.g., the Olympus XZ looks like a Canon Powershot S. But the XZ's lens is 28-112mm eq. f/1.8-2.5. The most current Powershot S's lens is a 24-120mm eq f/1.8-5.7. That makes a difference for low light, zoomed in shots.

Sensor technology/age

The other big difference that can show up in image quality is sensor technology. While two cameras may both be small compacts, an older model will nearly always have an older sensor with worse high iso noise performance. My old Canon Powershot S30 had (for me) unacceptable noise above iso 400. My Canon Powershot S90 worked well for me up to iso 1600. I'd posit that today's S120 is even better. Knowing the age of the model is key. Not all the currently offered models are necessarily the latest model, and if you're shopping used, this can make a huge difference between what spec out as nominally similar cameras.