Fewer megapixels
To get more pixels on a tiny sensor, the pixels have to be smaller, which creates more noise and poorer dynamic range. To combat these things, the camera needs to do more image processing, which is not good for photo quality. On top of this, the actual file sizes are huge or heavily compressed with the latter potentially causing more image quality problems.
This is specsmanship at its worst and really needs to stop. And while I'm at it, the manufacturers should cease talking about digital zoom like its a usable feature.
Larger sensors
The biggest hurdle here is price, because manufacturing a sensor that's twice the size, well, doubles the cost. Not good when you're trying to sell cheaper and cheaper cameras. As it stands, if you want the best photo quality you're going to have to pay for it.