"cosmos" is a countable word, normally used with "the", because there is only one cosmos in the real world. It is a synonym of "universe". "space" is the uncountable word to describe everything outside Earth.

Personally I would never use the word cosmos, I would always say space. I associate cosmos with one specific context: Russian space exploration (as opposed to American space exploration) - Russian astronauts are called cosmonauts.

In addition to Michael's answer, "space" tends to be used for things relatively close to Earth. The planets are out in space.bits also a less technical term "Cosmos" has the feeling of what is often called "deep space", covering objects much further from Earth. It's more technical and is often used when talking about the science and physics of the objects.