The size and Rivers can have a branch where Creeks don't.
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A River is a collection of Streams, a Stream doesn't have any other streams running into it and exists by itself. Rivers are usually larger than streams, though not always. The Nile and the Amazon are two of the largest rivers in the world, but there are many rivers that are much smaller, and sometimes even rivers that "vanish" for part of their length (they run underground for part of the distance)!
Officially, there is no difference between a river and a stream. The Geographic Names Information System classifies all "linear flowing bodies of water" as streams. Over 120 naming terms fall into the category of streams. Basically the only way to differentiate between a river and a stream is in the name given to the body of water.
Usually it's simply a matter of opinion. A stream is a smaller version of a river, but one countries river can be another countries stream. A basic example being - USA has thousands of waterways called streams / creeks that are larger than most of Japans rivers.
All rivers are streams but not all streams are rivers. There is a defined method (Link) of sizing flowing water, all these features are streams and are ranked as 1st through 12th order streams. The systems starts with what would be called in non-scientific terms brooks and works up to rivers. Going up in size and strength, streams that are classified as 4th through 6th order are medium streams while anything larger is a river. Examples:
a river and a stream are both fast moving bodies of water but the river is called a river because it is larger, deep, longer and wider as a stream is a thin shallow where you can walk across
The difference between a river and a stream is a river is deep and a stream is narrow and deep