My fantasy world consists of a flat plane, with the sun affixed to a point directly over the main continent. This sun powers down to create night conditions, and then powers up again in the morning. Given that it will never change its position in the sky, what are some likely ways this environment will differ from our own?
You would have absolutely no seasons whatsoever, and thus - depending on the output of your star - either a lush evergreen jungle or a desert, or anything in between, but no changing environment that has different behavior depending on seasons.
Also, lack of moonlight being reflected means pitch dark nights, which would influence predators. Better night vision is necessary, while other species might prefer to stay in sheltered locations during the night. Fire (or another light source) will be very important for an intelligent species that does not have night vision.
Sun dials as clocks will not be possible, so time has to be kept track of in another way. Similarly, animals need to have some kind of "sixth sense" to tell time and to go to shelter in time. How they do that without any apparent way to tell the time? No idea, honestly.
Plantlife would probably develop differently, since trees will now completely block the light. So no plants underneath trees anymore (with a rising/setting sun, those get at least some sunlight, if the sun is fixed, they do not).
For an intelligent species, navigation would be difficult without a way to tell cardinal directions by watching the sun.
There will be lots of small but noticeable differences, listing them all would prove futile.
The ecosystem on tree trunks will be remarkable. Moss (and perhaps lichens) will grow in two vertical bands (where they have some light but are not too blazed by the sun.). There may be fungi that grow in the dark on the far side of the tree, and perhaps up the trunk a bit (though that may not be true at all if the tree is healthy.) The leaves on the tree will be smaller and more curled towards the sun and broader and more flat in the shade. (Assuming an average climate.)
There will be effects on lakes. Assuming average rainfall throughout your world, those on the sunny sides of the mountains will evaporate more quickly than those on the shady side.
I hope these ideas are useful. Enjoy thinking about the small denizens of your world.
Due to the Earth being a sphere, the equator gets direct sunlight while the arctic regions see the sunlight from an angle. On a flat world, I would suspect the amount of energy the Earth is receiving would be pretty uniform across the entire landmass. From that point of view, I don't believe you would get an arctic/antarctic forming and instead have much more uniform conditions north to south. Instead of latitude being the big difference maker, Albedo becomes the major differences in terms of temperature (IE the ocean absorbs most of the suns rays while land tends to reflect it more).
Much of weather is ultimately the uneven heating of the globe...with more uniform heating/cooling, one would expect weather patterns to be weaker and as such more erratic. To compound this, much of weather patterns is heavily effected by the rotation of the globe...without a rotation, the weather would be even more erratic day to day. I'd expect local thunderstorms (sun heats up land, evaporates huge amounts of water, and drops it in a storm) is going to be the may stay of your weather patterns.
The planet will lack a tide as well, which will have an impact on some parts of life and it's hard to speculate on what ocean currents (if any) will come from this (earths currents are heavily effected by the waters warming at the equator and cooling at the poles...would that happen on a flat world with a fixed sun?). Air currents on Earth transports clouds (ultimately rain) from over the ocean to land...would this flat world see that or will all the oceans evaporated rain simply dump back into the ocean?...if that's true you would see a lot of the inner landmass as desert as no air currents exist to bring the rain clouds over.
Oddly enough, from a climate point of view, the flat planet plus no rotation actually has a heavier impact on the environment than the fixed sun position.
The difference probably wouldn't be huge. I'm assuming that things like gravity and atmosphere function the same way in your world as they do in ours. The only actual difference would be that the light in your would always comes from the same direction.
Do keep in mind that you will also have to create some sort of moon or other method of creating light during nighttime, unless you want it to be completely dark (which would greatly impact the environment).
That being said, the only changes would be some very specific things, like sunflowers or similar plants, which would always face in the same direction.
Overall there would be no major difference.