Biomes and ecosystems are both levels of ecological organization. A biome is a conglomeration of ecosystems. Thus, biomes will share all the characteristics of the ecosystems that comprise it. However, an ecosystem may not share all the characteristics of the biome that it is a part of.
A biome is a very large geographical area that is characterized by its climate, geographical features, fauna, and vegetation.
An ecosystem is a group of different populations and the interactions of those populations with their environment. In other words, the interaction of several populations with one another and the abiotic features within their habitats comprise an ecosystem.
Thus, biomes are much larger than ecosystems. However, both biomes and ecosystems both involve many organisms and the interaction of these organisms with their environment(s).