A stream is just a small river. A river is a large stream :) Note that "stream" (but not river) can also be used as a verb, meaning to flow continuously in a certain direction.

Generally speaking: Rivers usually have names. Streams usually don't. Rivers are more permanent. They run all year and change their path only slowly. Streams often flow temporarily or seasonally. Rivers run along the valley floor. Streams run down a slope and end at a river.

A stream is just a small river - I'm not aware of any "official" difference, as there are in some languages. We have lots of words for small rivers: stream, rivulet, burn etc. etc.

A stream is a small river. I would say streams are shallower than rivers too.