First off, this is a very cool concept! Your question is fairly broad, so I was going to answer a few different ways.
Society - just like how oceans and mountains separate societies and cultures in our world, the separate floating continents would likely give rise to different cultures. This would extend to their belief systems (do they all know/believe the gods raised the continents or do they each develop different beliefs about how/why floating lands exist? What legends have they come up with about the Earth's core?) and their technology (presumably, some societies would invent aerial vehicles to travel around while others might exploit natural resources - such as huge flying birds).
Biology - depending on how high up these continents are, your world's inhabitants would have adapted to slightly less gravity (though the difference is probably negligible) AND less oxygen. To be more precise, the concentration of oxygen is the same whether you are standing at sea level or on Mt. Everest (21%) but there is less atmospheric pressure, so it is harder to breathe. 279
If your societies are human, their bodies would have adapted by breathing faster and producing more hemoglobin. Because of this, if the humans were to descend to the Earth's "surface", their stamina would likely increase since they are now presented with "more" oxygen for their bodies to use.
Economy - I imagine mining would be a great deal easier with so much of the underground earth exposed, so you might want to think of how a society would look with an abundance of precious metals. Would their currency be based on something entirely different from gold and silver?
Water - though it would still be replenished through rain, having rivers that flow off the continents would pretty quickly lead to a water shortage. So either water is very precious and/or they have much more advanced ways of saving water.
Land - earthquakes wouldn't exist, right? Is there a concern the continents might run into each other from time to time? Do they have technology to deal with this?
It seems to me that there is a lot you could do with this concept. Good luck! 693