It could be argued that the distinction between the Holy Roman Empire based in what is now France, Germany, and Italy and that of the "Byzantine" Empire in Constantinople began when Charlemagne was crowned Emperor by Pope Leo III in 800. Empress Irene had taken over for her son, Constantine VI, who had been deposed and blinded 3 years earlier by her supporters. The Pope was propping up Charlemagne as the rightful Roman Emperor, and disregarding Irene's role as illegitimate.

Of course, the dissonance between the Greek speaking Eastern Empire and the Latin speaking Western Empire was strong and growing even in Late Antiquity. The issue here is that Charlemagne did not see the HRE as a rival state to the Byzantine empire; he considered himself the legitimate heir of the Caesars, thus part of his imperial title being 'Karolus serenissimus Augustus' meaning Karl most serene Augustus.

Their relations varied over time, but the 10th century travelogue of Liutprand of Cremona makes it very clear that Westerners and Easterners found one another weird and kind of gross on a personal level. The Easterners were very haughty and practiced a somewhat different flavor of Christianity from the Westerners (even though the churches were technically still united until 1054). Liutprand complains about the food, the intricate court etiquette, and just about everything else he encounters in Relatio de legatione Constantinopolitana ad Nicephorum Phocam.

I'm assuming the Byzantine Empire didn't really like the Holy Roman Empire, who claim to be the successor to the Western Roman Empire

They claimed to be the successor of the Roman Empire. There was no "Western" and "Eastern" Roman Empire before Charlemagne was crowned, just the offices of the Emperor of the East and the Emperor of the West.

The Pope considered Charlemagne the successor of Constantine VI, not Romulus Augustus or Julius Nepos.