The eastern Roman empire is the same eastern Roman empire as that when the western roman empire fell. Whereas the Holy Roman Empire was founded by Charlemagne who was Frankish, which is like French, and didn't even include italy to begin with. Not to mention the HRE wasn't even Imperial despite that being like Rome's thing. Not to mention the HRE is legitimised by the Pope who the Emperor supposedly rules over in the old Roman empire. Much like how the Baselius has control over the Ecumenical Patriarch of Constantinople.
Meanwhile, Byzantium kept many Roman traditions like, being imperial, retaining the Roman senate (in some form), referred to the emperor was Caesar Augustus (until some other emperor down the line changed it to Baselius), chariot races and gladiators in the hippodrome/Colosseum and being ruled by laws not kings (Roman law, and Justinian's code).
Ultimately the HRE is considered the successor of Rome not Rome itself. While Byzantium is Rome but with half of it bitten off.
correct me if I'm wrong.