The most important thing to note in differentiating ecosystem from biome is that an ecosystem or many ecosystems are part of larger biomes. This means that biomes are the larger categories of ecological units while ecosystems are subcategories. While there are five planetary biomes, there are many and varied ecosystems and each biome has multiple (perhaps myriad) ecosystems within it.
A biome, of which there are five major types, is a community of biotic plants and animals is a region that shares the same weather and temperature [there is actually debate as to whether "biome" should also include abiotic factors (non-biological factors) in its definition]. An example of a biome that is realtively easy to visualize is a desert biome. In a desert biome, which can cover vast expanses of area, the region shares the same weather, temperature, animals and plants.
The five major biomes--which are divided into water and terrestrial subsets--are forest, desert, aquatic, tundra, and grasslands. The acquatic biome has a number of sub-systems: fresh water, estuary, coral reef, marine (ocean), and wetlands. The terrestrial biomes also have sub-systems, like savanah and alpine. Within each biome and the sub-systems of the five major biomes are multiple ecosystems.
Ecosystems are communities of biotic and abiotic factors--meaning plants, animals detritus, minerals, microorganisms, rocks, soil, water and environmental climate and weather--that interact togeher. These biotic and abiotic factors all interact with each other in such systems as the hydrological cycle, the carbon cycle, the nitrogen cycle and the food chain in which solar energy energizes primary producers that are consumed by secondary producers/consumers and tertiary consumers all of which are eventually decomposed by detritus in the decomposition cycle.
Ecosystems may be as complex as an estuary or a wetland system or as simple as a single pond. Thus every biome has within it--within the limits of its weather, climate, and adapted plants and animals--manifold individual ecosystems, each of which has its own community of interaction.