Hmm... I've always known his full name, but it never really bugged me before that he's not commonly referred to as "van Beethoven" rather than simply 'Beethoven'. It's always "Beethoven's n<sup>th</sup> Symphony" instead of "van Beethoven's n<sup>th</sup>", which is actually quite odd and inconsistent with how Germanic names are usually treated in English.
Von is the German equivalent of van, and has the same meaning in this context. His parents were named van Beethoven, so it's not an affectation on his part - and according to Wikipedia, 'van' isn't associated with noble names in the same way as 'von'. They mean the same thing, but the usage varies.
If he's called 'Ludwig von Beethoven' in a source, then presumably it's either an error, or a German text that's using the German equivalent of his name, or it's simply using the German word for whatever reason.
My real name is The von CeejThat doesn't make a whole lot of sense - 'The of Ceej'? 'The from Ceej'? It only really works with proper names, I believe.