Ecoregions, or Ecological Systems, or Ecosystems, or Biomes, are defined differently for different purposes. Many scientists have attempted to develop objective classifications for ecosystems modeled after Linnaeus' classification for species; however, there remains no standardized way for talking about the bio/geo landforms of the Earth.
"Biomes" usually refer to the largest physiogonmic classes of vegetation on the earth, such as Tundra, Conifer Forest, Deciduous Forest, Tropical Forest, Grassland, and Desert. They are determined by climate. Classifying nature can be useful to determine the relative effect of man's activities in different places. For example, here is a map of America's Army Bases and biomes, color-coded to show where military training exercises can be most damaging to the environment. Restorationists use ecological classifications to determine what a particular landscape used to look like, and hence what seeds to buy. Native Seed Network uses EPA's level III map of ecoregions, but they are aware that this is just one subjective mapping effort among many.
For example, a glance at the following biome maps of North America shows some of the differences.
Maps of the U.S.:
Zooming in even more can help explore the differences. Maps of Western Washington: