I know that the Byzantine Empire was technically the Eastern Roman Empire, but did they carry over actual Roman traditions with them? Also, did the Holy Roman Empire have anything to do with the original Roman Empire, because I feel like since they were more Germanic, they weren't the same. So, were there any legitimate empires that succeeded the Roman Empire, and if so who?
Andrew Wheatcroft's The Enemy at the Gate: Habsburgs, Ottomans, and the Battle for Europe contains the argument that both the Habsburg Empire (in the West) and the Ottoman Empire (in the East) considered themselves to be successors of the Roman Empire by the 17th century.
Wheatcroft points out that both empires ascended almost in parallel: Frederick III became the first Habsburg Holy Roman Emperor in 1452, Mehmet II conquered Constantinople in 1483. (They also exited finally in parallel after WW I.) In this author's opinion they were bound to clash once the intermediate Kingdom of Hungary had declined, which was the case e.g. by the late 17th century.
As for the Habsburg Empire, its ruler bore the title Holy Roman Emperor (also King of Jerusalem, among many others) and it was variously connected to the papacy in Rome. (Wheatcroft's account specifically argues that pope Innocent XI contributed vital funds, without which the Battle of Vienna may have ended in Habsburg's disfavor. John Stoye's account of the same historical event argues that the pope's diplomats may have been instrumental in brokering the alliance between the Emperor and King Sobieski of Poland, which was also clearly vital for the Habsburgs in 1683.) On the other hand, Voltaire's quip is also instructive as to whether the Holy Roman Empire had "actually" anything to do with the Roman Empire:
As for the Ottoman Empire, its capital was Constantinople, obviously the last bastion of the (Eastern) Roman Empire but with rulers ultimately from the central Asian plains.
IHO due to their sustained military power over several centuries and their (somewhat legitimate) claims these two empires had somewhat (relatively) better claims than other pretenders as "successors of the Roman Empire", of which there must obviously have been (perhaps still will be) many in history.
P. S.: It's nice: I've now encountered already several (1, 2, 3) reasons and opportunities to cite from a book I've been reading recently on the Battle of Vienna and its historic embedding. A conversation (4) on History StackExchange triggered my reading of that book in the first place, and it was a good read too :-)
The eastern Roman empire was pretty different culturally by the time Rome fell - much more Greek obviosuly
No - it was just advertising. The Roman empire was pretty great, therefore if I want people to think I'm great I'll call myself the continuation of Rome. Throw in the holy to keep the church on your side
It might be slightly uncharitable to consider the Catholic church to be the continuation of the roman empire - but the last incumbent looked distinctly imperial.