The semicolon is indistinguishable from a full stop in speech, i.e, language. Hence its use is purely stylistic, applicable to the technology of writing, not language. And certainly not grammar. In writing a semicolon is a handy piece of artifice that can be made to serve a writer's purpose, like any other tool.
One of the purposes a writer may have is distinguishing nested lists that would be easy to understand by intonation and rhythm in speech, by representing certain comma intonations with a semicolon, as in the article that Armen posted. But mostly they're used the way Akin suggests in the question.
As Lewis Thomas put it,
I have grown fond of semicolons in recent years. The semicolon tells you that there is still some question about the preceding full sentence; something needs to be added; it reminds you sometimes of the Greek usage. It is almost always a greater pleasure to come across a semicolon than a period. The period tells you that that is that; if you didn't get all the meaning you wanted or expected, anyway you got all the writer intended to parcel out and now you have to move along. But with a semicolon there you get a pleasant little feeling of expectancy; there is more to come; to read on; it will get clearer.