I feel like the two situations are pretty much the same. In both cases is an empire that calls itself Roman but isn't what we generally think of as Roman (granted the Byzantines are in fact the direct continuation of Rome). The Byzantines called themselves Roman, and the name Byzantine was simply applied after the fact by historians to distinguish them from what one might call the "true" Roman Empire. So why is it then that they don't take the same approach with the Holy Roman Empire? Call it Germany, or perhaps an original name? Unlike the Byzantine Empire, they have no actual common ancestry to the Roman Empire, with their Emperor being of a completely unrelated origin, so it seems reasonable to think they might need a name to distinguish them from the actual Roman Empire.
We actually call it Germany but depends of the culture I guess.
I'm Italian and here the first time we use the word Holy Roman Empire is to talk about Charlemagne, in Italian Sacro Romano Impero. When we talk about the Empire born in 962 with Otto I we call it Holy Roman - German - Empire, in Italian Sacro Romano Impero Germanico.
Is not uncommon for us, especially in academia to refer to the Emperors of the Empire from Otto I to Francis II as Emperors of Germany or Emperors of the West since "Imperatore del Sacro Romano Impero Germanico" is too long. At the same time, Byzantine Emperor, East Emperor and Roman Emperor as used interchangeably.
I am aware that similar to us I know that in Spain they refer to the HRI as just Germany. This can be seen when they talk about Charles V which they call "Carlos I de España y V de Alemania" which translates to Charles I of Spain and V of Germany.
I suppose the English word never had the need to differentiate between two empires since the Eastern one was so far away and therefor they stayed with one title to call the German one and adopted Byzantine as a way to call the Greek Empire after it fell, while the Mediterranean world that was influenced by both had the need to differentiate between two Empires.
Sorry if I made some typos.