Some anthropological knowledge may be lost, Britain has one fewer Russophobic political commentator. I'm not sure much else would change, though. He was one voice among many, politically, and I'm not too familiar with his cultural work.
I think that the perception of the Ottoman Empire in Hungary would be significantly less favorable, with a popular culture that does not emphasize the "steppe" bit of Magyar ethnogenesis as much. Not sure how it would impact things in the broader picture. Maybe the treatment of Bosniak Muslims under Austro-Hungarian administration would be less relatively "amicable" allowing them to find more of a common ground with local Serbs. That would be potentially huge.
More realistically, however, the general train of thought that Vambery supported would likely find other advocates. A generally somewhat positive view of the Ottomans and Islam in Hungary is likely to emerge anyway because of political trends of the era.
Ignaz Goldziher could be more influential on the cultivated general public.