The Roman Kingdom/Republic/Empire is a really fascinating period of ancient history. What's really mindblowing is when you realize that the Eastern (Byzantine) Empire didn't fall until 1453.
If you're looking for interesting listening on Rome, I highly recommend Dan Carlin's Hardcore History Podcast. The series Death Throes of the Republic focuses on the period where Rome transformed from a Republic to an Empire (maps 13-16). I just finished listening to that a few weeks ago, and it was amazing. Additionally, Thor's Hammer discusses what happened in Europe when Rome fell. There's also an older series on the Punic wars, but I haven't had a chance to listen to that one yet.
I love reading the book of Acts in the Bible because it relates their travels within a part of the Roman empire. What strikes me is how similar to modern day traveling it seemed to be, just in longer timescales. But the same way we could say today "I'll catch a plane to Madrid and from there I will proceed to Paris", you read there about people taking ships and then traveling on foot to other locations as if it weren't such a huge deal i.e. not as if they were saying "I'm embarking on this long journey to the end of the world, farewell!"
I guess we could say the part of the world covered by the empire was quite "globalized" at that time, in a sense.
Christianity wouldn't have spread nearly as quickly or as far if it hadn't been able to piggyback on the Roman Empire's infrastructure. On the flip side, after Rome's political influence left most of Europe, the parallel church structure was still there to help maintain civilization at least a little bit during the barbarian invasions and general collapse.
[Leaving out Mithraism, which also spread in similar ways, as well as any discussion of what exactly constitutes "barbarian" vs. "civilization" since it gets fuzzier the closer you look.]
The soldiers across the Empire came from all the imaginable parts of it. Going through the museums in Europe near to the former Roman camps, I've seen the statues of Egyptian gods brought or made by Egyptian-born soldiers or artisans, found between the statues of Mithra.
The Pax Romana was probably a big part of that. During that time a person could reasonably expect to travel through most of the Roman Empire without running into anything worse than local thieves and without having to deal with national borders.
Both before and after that time period in Western Europe, in contrast, any long-distance travel was likely to run into wars of expansion or civil wars, local warlords, or so on.
Another good podcast on the subject is Lars Brownworth's 12 Byzantine Rulers (http://12byzantinerulers.com/), which starts with the Roman emperor Diocletian and proceeds through the founding of Constantinople, the fall of the western empire, and the evolution of the eastern empire over the 1,000 years that followed.
Mike Duncan has since moved on to Revolutions and is currently in the beginning of the French revolution. Don't let that slow you down, its not like the history of rome is going to change much in the near future.
Then "the history of rome" podcast begat "the history of byzantium" podcast. Another investment in time which isn't nearly done yet.
There is also the British History podcast which is pretty good listening and technically on topic as it was partially part of the roman empire for a good while. Another work in progress.
(And I see sspiff and I posted within seconds of each other. Well, great minds think alike and all that...)
He wrote a book on the eastern "Byzantine" empire as well, Lost to the West, which covers some of the same ground as the podcast series but plenty of other material as well.
Also by Lars -- Norman Centuries -- fascinating account of how Vikings ended up ruling Britain, Normandy, Sicily and leading crusades. http://normancenturies.com/
Just finishing up the Fall of the Roman Republic now, it's a fantastic series and podcast. Unrelated to Rome: Dan Carlin's Hardcore History series on the Mongols is equally awesome.
There is this TV show "Ancient Impossible" which showcases all kinds of awesome things that people built thousands of years ago. The achievements of Romans (along with Egyptians) are the most awe inspiring
More distinct, definitely. But the interesting bit is that they themselves would never have recognized this. Walk up to someone calling themselves "Roman" in 1 AD and you could find yourself talking to a pagan born in modern Italy who spoke Latin. Walk up to someone calling themselves "Roman" 500 years later and you could be talking to a Christian born in modern Turkey who spoke Greek, which is a very very different person. But both of them would have insisted that they were "Roman." (The idea of "Byzantium" itself as something distinct from the Roman Empire is a purely modern construction -- no Byzantine would have conceived of themselves as such.) So the idea of "Romanness" was socially constructed; it meant whatever people living in the empire at a particular time collectively decided it meant.
You can see a similar type of socially constructed identity by looking at "whiteness" in American culture. 150 years ago, a "white American" was strictly interpreted as a person of northern European ("Anglo-Saxon") descent. English, French, Germans and Scandinavians were "white people"; Spaniards, Italians, Irish, Poles, Russians and Jews were not. But fast-forward to today, and all those latter ethnicities are now commonly identified as "white" in American culture.
In other words, whiteness, like Romanness, is less a fixed identifier of a particular ethnic/religious/linguistic configuration and more a convenient way for people to divide the world into in-groups and out-groups, "us" and "them."
By the medieval period, vastly different. There are stories about when the nobles of the first Crusade visited Constantinople they were thinking how much different the Emperors of the East were compared to what they knew of the old Roman Empire. As mentioned before the East was always different from the Western empire, but they really drifted apart in the 400's.
The problem with the question is that it simple depends. So first, even within the roman empire there was change. A Roman from 500 b.c. would not think 500 a.d. was very roman. A roman from 500 a.d. would not think 1400 a.d. is very roman.
So the answer to your question really is, in the begginning they where the same and then slowly shifted away from it. What can be said is that there is no clear break, no point where we can clearly say "here" now its something diffrent. This is why we have so many problems in nailing down the end of the roman empire or the begginning of the byzantine empire.
So I think, the only sencable thing to do is to still call them romans (or rather the roman empire).
More like north and south carolina, at least until things went sharply downhill for the western half.
Or perhaps a better analogy is its almost exactly like the U.P. of Michigan vs the mitten of Michigan (lower michigan?)
One problem with discussing how close two political orgs are over two thousand years is times change. Relations definitely had their ups and downs over the whole course of time. At times the religious strife between west and east was a little fiercer than anything that happened between US and Britain, at least post-1812.
(edited to add in a really good familiar analogy, at least some of the time ... Britain and Ireland)
Since Latin was the language of the western empire, but Greek the language of the eastern empire, I'd say that the two halves were more distinct that the U.S. and the U.K. are.
Death Throes of the Republic is my favorite podcast series ever and I've listened to a lot (having a braindead job where I would get away with just staring into a screen listening to podcasts all day long).
If you're looking for interesting listening on Rome, I highly recommend Dan Carlin's Hardcore History Podcast. The series Death Throes of the Republic focuses on the period where Rome transformed from a Republic to an Empire (maps 13-16). I just finished listening to that a few weeks ago, and it was amazing. Additionally, Thor's Hammer discusses what happened in Europe when Rome fell. There's also an older series on the Punic wars, but I haven't had a chance to listen to that one yet.