An independent clause is a statement that could have been its own sentence. Funnybone is saying that you use it instead of a period when the two parts of the sentence are complete statements which you decided to make into one sentence.
A long, run-on sentence should have commas in the right places to show phrasing. However, when you will need sub-commas, as in a parenthetical statement or explanation, within and in between more important commas, you might want to use the semi-colon. Otherwise, it may confuse the reader, who will have to decipher which commas are serving what purpose.
So, it is less than a period and more than a comma; it can replace a comma and resemble a period.
When you are listing Rabbonim in a picture, you’ll say: R’ Asher, Rav of Khal Bnei Zelda; R’ Berel; Mashgiash of Yeshivas Ahavas Mussar. If you’d use commas throughout, you would think that R’ Berel is the Mashgiach of Ahavas Mussar. I know. I could have used the pipe. ;|