Why is Vincent Van Gogh known as Van Gogh, but Ludwig Van Beethoven known as Beethoven? Is it a convention of nationality, or something grammatical?

I suspect that this has simply become a convention, possibly having something to do with the length of Ludwig's surname vs. Vincent's. In Art History, we never call van Gogh "Gogh," but I may have to call upon some specialists in Dutch art in order to figure out exactly why not.

https://en.wikipedia.org/wiki/Life_and_work_of_Ludwig_van_Beethoven

Prior to the late 15th century or so, surnames were relatively rare among Europeans, unless they were members of prominent families (In Italy, the Crescenzi, the Colonna, and eventually the Medici.)

Most ordinary surnames were originally derived from place names ("Leonardo da Vinci," is a good example), personal characteristics, professions, and father's names. And sometimes from genitive prepositions combined with these places, nouns, and adjectives. If you meet an Italian whose surname is "DiStefano," then you can be sure that somewhere back in his history, he had an ancestor whose father's name was "Stefano." Or, if his name is "Di Rossi," he probably came from a family full of red-headed people.

"Di" (also "del" and "della") in Italian, and "de" in French are the grammatical equivalents of "van" and "von." Over time, the genitive prepositions became attached to certain names; in other cases, they were dropped. What happened in each case was a matter of individual history and fortune.

Here's a bit of one family's history that goes part of the way towards explaining it:

"Basically, people in Holland in the olden days did not have a surname; they were known only by a first name and a point of reference - like from someplace, or someone's son. This system changed in Holland when a system of registration was started by Napoleon, and this explained the variances in the spelling of many families, including the Valkenburgs. A similar process happened earlier when settlers came to the USA and needed to have a more concrete identity than, say, Lambert. So Lambert could have referred to himself as Lambert van Valkenburgh (Lambert from Valkenburg)."

"And, theoretically, these six families and their descendants are entitled to use the "Van" between their Christian and Surnames. I say theoretically because of course over the centuries some people entitled to the Van dropped it, or had it dropped during Napoleon's registration process, and others used it in emigrating to other countries, which had a system of recording names."

https://www.familysearch.org/wiki/en/Netherlands_-_Dutch_Research_Introduction_(National_Institute)

That spelling issue with Beethoven (spelling it "von") actually became a critical point in a legal battle he was fighting for the custody of his nephew when his brother died.

I think the painter's name is pronounced "van Gogh" simply because his last name's just too short. With Beethoven that's not a problem (even if you adopt Stanley Kubrick's old habit of referring to him as "Ludwig Van").

Van Gogh is Dutch, Beethoven is German. It is Dutch custom to include Van as part of the family name if formally used in the name. E.g., Vermeer is just Vermeer, Erasmus is Erasmus, but with Van Halen and Van Gogh, the Van is part of their family name. It is similar to O'Connell and Connell in Irish. Ooh, Beethoven is German and the van is a kind of prefix but is not part of the name.