Here are just some examples.
Low light photography
You say that your main concern is low light photography. This is actually an area which will show a significant difference.
On a continuum between very small sensors and large sensors, cellphone cameras are at the very bottom of the pile. Compact cameras have pretty small sensors, so they too have a reputation for poor low light photography, but they are usually not as small as those on a smartphone or other cellphone.
The effect of a small sensor size is that, all else being equal, low light photography will have higher noise levels. Since modern cameras try to filter out the noise, it will mean that low light photos will generally become more "muddy".
As you mentioned, cellphone cameras tend to have a fixed focal length and can only zoom digitally, which degrades effective resolution as you zoom in. Some compact cameras now have incredibly long optical zoom lengths, eg 14x which is the equivalent of 28-400mm on an old full frame camera. They can include these long zooms without too much cost or bulk because of the small sensor size.
Depth of field
The smaller the sensor and lenses, the wider the depth of field, all else being equal.
This is both an advantage and disadvantage of smaller sensor sizes: the advantage being that you can get more in focus with a wider depth of field, but the disadvantage that it's harder to get less in focus: ie, to blur the background.
Due to their slightly larger size, compact cameras will be a little better at this than cellphone cameras.