Is there some official point of measure (size? volume of water?) that separates a stream and a river? Some stage where a river can definately be said to be a river and not a run, brook, or creek?
Someone may know an official guideline, from the U.S.G.S. for example. But it’s my distinct impression that it’s arbitrary – the larger streams in a given region are called rivers and the smaller creeks, brooks, branches, etc., regardless of how they compare on a national basis.
(It’s also worth remembering that the headwaters of a river are still usually considered part of that river, even though they may be a very small stream. I’ve personally waded across the Mohawk River, and once jumped across it (two jumps, with a convenient rock) – which would surprise Schenectady and Albany Dopers where it’s a major waterway. I was in a rural Lewis County (NY) town near its source at the time. I suspect a few Minnesota Dopers have done likewise to the Mississippi, just to say they’ve done it.)
I’ve always been taught that a stream is very broad. Basically a stream is all the water that flows within a watershed. So that includes the major river(s) and all the associated tributaries (some would even go as far as including rainfall) in a given geographic area. A stream ecologist studies the biology associated with moving freshwater (as opposed to stuff in ponds and lakes). So a stream isn’t just the little brook behind your house.
Of course, when everyday people use the word “stream”, they mean a brook or a channel or small river. And like the terms “brook”, “channel”, or “small river”, everyone has their own standard.
Okay, so a gully is a little run that is a gully whether it’s wet or dry, and a constant flow of water, no matter the size, is a stream.
Thanks, guys.
Sorry, it’s not that easy. It’s local nomenclature. I travel all over Montana and fish and float and what is a creek in one place is a river in another.
One measure of volume moving down a river/stream/creek is cfs - cubic feet per second. This is the amount of water that flows by a given point in a second.
For example, Rock Creek in western Montana has an average streamflow of 2024 cubic feet per second (cfs) on June 1 near where it flows into the Clark Fork river.
The Tongue river in eastern Montana has an average streamflow of 1126 cubic feet per second near where it flows into the Yellowstone river.
Whistepig
Nearly all rivers were named long before there was any standardization.
In general, a stream is a generic term for a narrow watercourse. There are probably very few things named “Stream.” They are too unobtrusive to name. I have seen Brooks, though, which are probably indistinguishable as to what is a stream.
Creeks can be any size. Schoharie Creek in New York is what most people would call a river. Jockey Creek, Goose Creek, and Town Creek in my home town aren’t freshwater at all; they are saltwater estuaries (which is the original definition of "creek).
Similary, the East River in NYC isn’t a river; it’s a strait. Same for the Harlem River, I believe.
I’d question that that was the (only) original definition of ‘creek’ - the local (named) creek to me isn’t an estuary, but is an inlet some ten miles up a tidal river.
This EPA website suggests that rivers empty into larger bodies of water such as lakes or oceans, whereas streams do not. In other words, streams empty into other streams or into rivers, but not into lakes or oceans; if it empties into a lake or an ocean, it’s a river.
Wikipedia seems to suggest that if it’s navigable, it’s a river (but again, not the converse).
It seems to me that it’s mainly arbitrary: rivers are just large streams that people have chosen to call “river” instead of “stream”, “creek”, “brook”, or anything else. “Brook” has, or at least once had, a specific meaning.
What was that?
A “brook” is a small stream, easily forded or crossed, running between two larger bodies of water. In Scotland, “brook” is used to refer to a tidal water channel.
Also, the use of “creek” to refer to a tidal water channel is English and has not been followed in the United States.
And how do rills, rivulets, burns, becks, runs, washes, runnels, streamlets, and freshets fit into our precise nomenclature? 
Stream is the most generic term. A river is a stream that flows into a body of water.
Or flows into another river. Or dries up in the middle of a desert.