The Jutes settled in SE England. The Vikings who came to England came from modern-day Denmark.
However, they seemed to have been not aware that they are the same poeple, and that they've only been removed from one another for ~4 centuries. This is like an Argentinian visiting Spain, IMHO.
I believe that the Juts who came to England were probably upper-middleclass, because they were politically affiliated with the Romans. Moreover, they had the wealth and motivation to move to Modern-Day England.
However, the Vikings were peasant farmers looking for adventure and economic opportunity. Perhaps this was a source of bickering.
Did they know of each other's origins?
Could they communicate with each other?
What makes you think they were unaware? Why would they have preserved the story of the Beowulf, for instance, if they thought it was some other people's myth?
But there was no single 'Danish' people in 400 or in 800; it only became a consolidated kingdom later. The Beowulf itself also attests to that, if one looks at the very many subdivisions of Danes listed in it (Ring-Danes, Spear-Danes, North-Danes, South-Danes, West-Danes, Scyldings, etc, etc) You're dealing with tribes here, not nationalities in the present-day sense. Post migration, there was not much distinction made between Angles, Jutes and Saxons either, hence "Ænglisc" ("Anglish"/English) ended being used as a collective term for the Germanic language used, regardless if spoken by former Jutes, Angles or Saxons. But there was more than enough time for the Anglo-Saxons in Britain to develop their own distinct identity.
The question of whether Old English and Old Norse were mutually intelligible is the subject of some academic debate. It's not easy to quantify such a thing in any case - it can depends a lot on who is listening and how much attention they're paying. But there's no way they'd not notice the languages were closely related - many words were simply identical: man, hand, hus (house), or had changes that were fairly regular: dead, read (red), bread in Old English is dauðr, rauðr, brauð in Old Norse.
Saying they were the "same people" might be a bit of an exaggeration. /u/Platypuskeeper goes into some detail on the facts of language, and tribes rather than countries, but there's also the fact that as you say, these men were ~400 years removed from their "homeland" in Denmark. That's longer than the vast majority of settlers in the new world have been removed from their previous homelands, but you'd hardly call an average American "English" or "German." The culture of England by the 800s was fairly different than that of the Danes who were raiding and invading in the same period- the Anglo-Saxons had been Christianized for a time and were somewhat more integrated into "Christendom" than the pagan Vikings raiding their shores. The fact that they were the "same people" (which, again, is something of an overstatement after 4 centuries of separate development- about as much time as it had been since England or Gaul were "Roman," at that time) likely wouldn't have been relevant. There were inter-tribal conflicts in the pagan lands between groups far more similar to one another than the Vikings and the Anglo-Saxons.
I personally don't understand the gist of your question. By what appears to be your definition, the Roman civil wars, wars between the Anglo-Saxon kingdoms within England, the American revolution, the Korean War, the Vietnamese war, and countless other wars were between the same people.
Maybe you could clarify your question.
