Lots of people have asked your question. Here are some links to answers:

link 2
link 3
link 4
link 5

There's even an 'ecosystem and biome song' on youtube, but I'm not sending you the link, because the beginning didn't interest me.

Reading a bit of each link, I'd say that there's a lot of overlap between the definitions of ecosystems and biomes - that's in the last link. Ecosystems are generally thought of as being smaller than biomes, so that there are a number of ecosystems within one biome. One link said that biomes consider the non-biological parts of the environment, too [the 'abiotic' parts] such as the soil and the weather. I like that, but Wikipedia says ecosystems also include the non-living parts of their environment.. I think of ecosystems as being systems of plants and animals that live together and compete and cooperate for food and other resources. I think 'ecosystem' is an older word than 'biome', at least in terms of common scientific usage.

I recommend checking out Wikipedia.com for their descriptions of the 2 words, too.