What are a level designer's roles in the development of a game, what are an environment artist's, and how do they compare? Are they overlapping roles? Does a game development team typically have both roles?
It depends on the studio. There are two fundamental camps. The terms are ones I've used, but probably aren't common elsewhere.
"Dallas-style" level designers typically do everything, typically using some kind of engine that has heavy brush editing (i.e. quake based). Usually from layout, texturing, placing props, lighting, NPC placement, scripting, etc. "Environment artist" isn't really a position that exists. Usually the art department are involved with making textures, ancillary props, skyboxes, and occasionally some integrated hero props (that are usually first roughed out with brushes and then exported into a format where the artist can make something over top of it in something like Max). You'll occasionally do things like paintovers from concept artists to give the LDs some artistic inspiration for the final art passes. The LD is primarily the one responsible for maintaining performance. A lot time this is with older engines that require a bit more care and technical knowledge because these engines are portal based.
"California-style" level designers mostly do layout and scripting only. This is more common with engines that are based around art tools (i.e. the LDs are also using Max). After gameplay is proved out, the environment artists go in and do the rest. Texturing, props, particles, sometimes lighting, and pretty much anything that isn't gameplay specific is done by the artists. They usually work pretty close with the concept artists. The artists have a good amount of responsibility with performance in this mode as well, usually in the form of "keep this number in the green" for a given scene.
Fortunately, "Dallas style" level designing seems to be going away. Splitting up responsibilities means that you aren't putting as much on the shoulders of the LDs. Plus you have a higher theoretical artistic bar if your artists are the ones doing the "arting up".
Why is "Dallas style" still popular at all? Mainly due to the engines being used. Artists generally are averse to learning LD tools. They can be very comfortable with Max/Maya, but trying to get a not-very-technical artist to use something like Radiant can be a bit of a disaster. Some studios have solved this as a culture issue.
An environment artist would focus on building the ambient nature of the world, ideally intended to immerse the player. This would probably range from the colours of the sunset, to the placement of non-gameplay oriented objects. A level designer would be implementing the placement of Ai, vehicles and necessary game objects.
If you think of a game such as Crysis, an environment artist would likely design the world in an artistic sense, then revise it based on changes made by the level designer after.
Contrary to what CommunistDuck mentioned, world objects such as doors and plants would probably be done by a 3d artist.
Level design is not really about visual design, although in many studios level designers do end up doing a lot of visual design work (eg. applying textures to the level.) Level design is really about planning the flow of the level, making decisions about gameplay, and laying out challenges for the player to overcome. Level design as a discipline starts with written plans and drawn schematics before firing up a level editing tool.
"Pure" level design is embodied by a workflow called "white-boxing" where the level designer places blank (not necessarily white, anymore than blueprints have to be blue) geometry in the scene to establish scale and structure, places all the dynamic items and enemies around the scene, and then environment artists go through and replace all the stand-in geometry with polished assets.
Here are a few links about game design and level design collected on my webpage: http://www.newarteest.com/game_dev.html
Especially take note of the link "evaluating game mechanics for depth" because that is a really informative article about level designers.