On very resource-poor maps, heavy expansion and cottage or hammer economies become crippled by city maintenance, since each individual city grows slowly, produces much less, etc. So certain alternative economies, for example the great lighthouse, become more useful as they contribute a larger percentage of your commerce per city. Also specialist economy and GS bulbs will be a higher percentage of your economy on poorer maps because you can do less traditional research.
Caveats: Reckless Expansion is pretty good regardless of map quality on difficulties below immortal since the maintenance is much lower
On very resource rich maps, we alter our strategy to claim as much of the richness as possible, as fast as possible, knowing that the cities will pay for themselves in short order. We can also take advantage of empire-wide benefits, like Mercantilism or the great lighthouse, because of this more aggressive expansion.
Another way of looking at it: as has already been said, poor maps are harder because the human is better at taking advantage of rich maps. On a poor map, we are fighting to win and must play superbly to take advantage of narrow opportunities to improve our positions. For example, this can be seen in lots of poor deity isolation maps played recently on the forums. Expansion, great people, tile improvements, everything are funneled into beelining astronomy, and later a narrow military advantage (usually cannons, cuirs, or rifles), then an army is rapidly whipped together from nothing to conquer some better land.
On the other hand, on a rich map, we are not necessarily fighting to win, more likely fighting to win quickly or convincingly. Where a poor map requires focus to do one thing well, a rich map requires vision and planning to realize exactly how much is possible. Conquer much of the world, expand rapidly, build the Pyramids, Oracle, and Great Lighthouse, all by 1000 BC? On a rich enough map, ridiculous things are possible (lots of HOF games are great examples of this, WastinTime's BC Space thread shows it especially well).