If something has been discovered in 1980 and I want to refer to this discovery in a scientific text, how would I do this?
"It has been shown, ..."
"It was shown, ..."
and especially: Why is one form used over the other?
*The present perfect - in this case a present perfect passive: "has shown" - is a form of present tense. It will always tell you something about the situation at the present time.
*The past tense - in this case a past simple passive: "was shown" - will always tell you something about the past. There is no connection with the present.
Let's look at your example:
If you do not specify when this happened, you can use a present perfect "It has been shown...". This tells us that we are now aware of whatever the discovery revealed, and that this discovery informs our current knowledge.
It's unusual to use the present perfect to talk about a discovery that took place decades ago, but it is possible. It suggests that the discovery has not been superseded by later discoveries.
However, if you say when the discovery took place, you have to use a past tense:
It was shown in 1980.
Note that this is an unbreakable rule - you can never use present perfect when the time is specified. This means that there is a basic grammatical error in your first sentence. We can say "something has been discovered" (without specifying the date) but we cannot say "something has been discovered in 1980". If you specify the date, you must use a past tense. This rule applies regardless of whether you are speaking or writing, in formal or informal English.
Mathias, please don't use this source http://www.differencebetween.net/language/difference-between-has-been-and-was/ or anything written by Nimisha Kaushik. What she says about English grammar is unhelpful and inaccurate.