What's the difference between the universe and the cosmos?

In technical (astrophysics) usage no.

There are a number of different uses of universe but cosmos isn't used as a specific technical term. Cosmos is sometimes used in popular works as a homage to Carl Sagan's famous TV series.

Cosmological is used as a technical term, eg. Cosmological constant, Cosmological redshift - because using "Universal" would be confusing.

Universe means "the whole world" or "all taken collectively".

While Cosmos comes from the Greek Kosmos (from the OED: κόσµος - order, ornament, world or universe (so called by Pythagoras or his disciples ‘from its perfect order and arrangement’).

Cosmos is the opposite of Chaos, which was the first state of the universe.

Nowadays they are used like synonyms; they refer to the same thing, but seen from different "point of views".

Cosmos means "the universe seen as a well-ordered whole."
Universe means "all existing matter and space considered as a whole; the cosmos."

The words can be used as synonym of each other, or you can use cosmos when you are referring to the well-ordered aspect of the universe.

"Multiverse" is the "container" you are referring to.

I believe that the "Cosmos" refers to the astronomical aspect of the Universe (stars and etc), whereas the "Universe" refers to the entire reality.