What is the difference between the words “stream”, “flow” etc.?

What is the difference in meaning and usage between the nouns “stream”, “flow” (“pour” is excluded from this list based on comments) and probably other words with similar meaning? What is more appropriate when talking about news or other entities of content continuously appearing on a website, other discrete object constantly appearing on one side, moving to other and disappearing there?

I would say that it is very uncommon to use the word flow in the context of:

"news (or other entities of content) continuously appearing on a website"

Stream in this context commonly refers to live video.

The word you are looking for as mentioned in the comments is feed — this is the commonly used noun for content continuously appearing on a website.

You're asking about a metaphor.

Before you think about the words, get the metaphor fixed in your mind.

• What is flowing? News? Words? Text?
• Is it incompressible like water or compressible like a gas?
• Where is it flowing from?
• Where is it flowing to?
• What is its channel?
• How much pressure does it have and where does it come from?
• Speed, volume, throughput, variability, tides/news cycle

Write up the metaphor frame -- a long description of the metaphor as you see it, but be literal. Use all the fluid verbs you can think of, but use a term for a fluid in the description, instead of news.

Then substitute news for the fluid and see which ones are cliches, which ones are just too weird, and which ones work. There'll be some of each for every metaphor frame.

For more about frames, see Framenet.