Which nation or people were the biggest threat to the Byzantine era?

Are there more than one?

The Persians certainly remained problematic for centuries, on and off. And they were descended and or associated with the older Parthians and Seleucids. The western folks just really have never learned how to get along with those neighbors even yet.

The Turks, Seljuk and Ottoman, certainly offered threat in the later years of the Byzantine era.

The Vandals were often ignored as they built themselves kingdoms in Spain and then North Africa--until they could no longer be ignored.

The Goths--were convenient--until they were no longer convenient.

Which is a worse threat--the enemy who simmers waiting behind his borders, until time comes again to wage battle and war, like the Persians, or the enemy who is at first welcomed, tolerated but then needs to be obliterated, like the Ostrogoths and then the Lombards thereafter?

Not easy to tell, because Eastern Empire existed more than 1000 years and during this age was different political situation in different centuries.

First: Persians (5-7 century). Byzantine inherited this nemesis after split of Roman Empire. Wars with Persians were bloody and finally noticeably weakened both empires in the worst possible time - during muslim rise.

Second: Muslims / Arabs (7-12 century). Muslims were in relatively easy situation. Byzantine empire was seriously weakened, Persians near to total collapse. After muslim attacks Persia definitely collapsed, Byzantine lost Levant and Egypt - important breadbasket. When Byzantine lost vast teritories, world power decreased to local power. From this time Byzantine wasn't not superpower, but only (but important) fortification against muslim invasion to Balkan.

Third: Ottomans (13-15 century). Although Ottomans destroyed Byzantine empire, they were mainly consequence of muslim wars (turkish people were islamized similar like Persians). In this time power of Byzantine was low, Empire was weak without possibility to reconquest lost teritories.

In my opinion Muslims / Arabs were the worst enemies.

Don't forget on orthodox - catholic relations. Bad relations were very important factor for stability East Roman Empire.

"Good" old enemies (like Goth, Vandals etc.) were secondary enemies. They were focused on western part of former Empire.