The things that matter on the demo screen are manufactured goods and crop yield. In reality, crop yield plus approval rating is a better way to gauge scientific competitiveness than is literacy rate. Manufactured goods is more important to consider when entering war than military is, though it is true the AI is foolish and doesn't use production well. Manufactured goods is also important because it tells you what buildings and wonders you can afford to denote production to. Early game, checking manufactured goods against GDP can tell what tiles your enemies are working in their capitals, and what kinds of things you can therefore afford to build. For example, if you are in a game with five others and have the highest production with 10 and the average at 6, then you know you're the only one working any production other than base. You can then cross reference this against GDP; if you are at 3 and the average is fairly higher, it can be assumed your enemies are working other tiles besides production.
And score is only important in the early game, to check who builds certain wonders, when others found their cities, who has a costal cap, etc. If your neighbor is lowest in military and production, it's an indicator as to what to focus on, as it is when your neighbor is highest on military and production.
Probably the significance of demographics ordered most important to least important are as follows:
Manufactured goods
Crop yield
Military
GDP
Land
Approval
Literacy
Population
Population and literacy are at the bottom because they are deceiving in that what they appear to measure is not what they actually measure; population does not tell actual population, and literacy has very little to do with measuring an opponent's actual practical science. Approval rating only has use in measuring when an opponent's growth will be curved, and land is not entirely relevant because typically if someone's actual size in terms of border growth is an issue then most likely you can see it from your own lands. Meanwhile, production, food, and gold reflect directly to the screen, and can be used to gauge how safe rushing wonders or settles is in comparison to opposing resource allocation. And military is just a temporary measurement of actual power, albeit a pretty important one to know when a sneak attack is coming.