First and foremost: Safety. Some of the cheapos are known to catch fire.
Other than that it's a sliding scale of part quality & reliability, how long the printer stays tuned after you tune it, what materials it can print (many cheap printers lose the heatbed, and lose quality in the hot-end so can only print the easier plastics), and other stuff.
Printers in that $200-$300 range often don't even
have a heated bed. That's something I run into with my Flashforge Finder. Differential cooling can cause the edges of a print to lift up off the stage, creating distortion. Depending on how bad, it can junk the part, but often the upper portions will print fine, but the bottom is spoon shaped. Heated bed helps prevent this and helps with adhesion of the initial layers. If those break loose, you come home from work and find a pile of PLA spaghetti in the machine. :-(
A lot of people need to be broken of the idea all prints need to be unitary/all in one go. Many can be broken up into logical chunks and assembled. There's a number of advantages, such as: bulk production of pieces; reduced chance and effect of print failure; better optimization of orientation; easier to paint/weather prior to assembly.
For example tall buildings could be printed as individual floors and stacked as needed for each game session.
This!! There are ways to take files you get from Thingiverse and edit them, but I don't know them yet. But presumably you are modeling your own parts. In that case, Tab A and Slot B are your friends. Dovetails are also wonderful. Just remember to add draft angles or it can bind up 1/16" too soon. Don't ask me how I know this. :-\
In general, a better machine, with higher resolution and thinner layers will mean less finishing work.
It probably won't be too big an issue for making terrain, but higher end machines have the option of printing support structures out of PVA. All machines should be able to do supports that you have to tear away later. PVA support is nice because it's water soluble. A warm water bath and a toothbrush are all you need to remove it, and it doesn't mar the surface the way homogenous supports will.
If you can afford it, avoid being pennywise and pound foolish in terms of build volume. Adhesion gets a bit squiffy at the edges of the the build space. My Flashforge officially has a 5"x5" table but I try
really hard to stay inside 4"x4".
