A biome is known to be larger than an ecosystem. In a biome, there may be different ecosystems that are available. For example, an aquatic biome may have some mini water forests that are available underwater as well as some coral reefs. The coral reefs and the mini water forests will be the ecosystems.
The ecosystem is usually defined as the interaction between living and non-living things. In a biome, the different living things are known to interact and live well within the environment. The different flowers that thrive and the different animals that live in the environment are considered to be part of the biome.