My interpretation is "plead something" has legal overtones and implies you're arguing or claiming as you might in a trial. "Plead the 5th" is sometimes used jokingly in conversation in the US to mean "I'm not going to say anything more" and is a reference to exercising your constitutional right under the 5th amendment to not incriminate yourself. It is related somewhat to legal pleading but it isn't always about a legal situation.
After my scholarship was unfairly revoked, I went to the provost to plead my case for it to be reinstated.
To "plead for" has less of a legal connotation and is more synonymous with "to beg or to implore".
Rodney King plead for an end to the rioting in Los Angeles when he said
People, I just want to say, can we all get along?