As I recall, GRRM's intention with Dunk & Egg was simply to write a novella set in Westeros, without creating any spoilers for ASOIAF.
In ASOIAF we have already learned the fate of Dunk & Egg (Ser Duncan the Tall and Aegon V), and that, I assume, is somewhat of a spoiler for certain people. So, I will elaborate a bit on that at the end, within spoiler tags.
I would say that as more time (our time =)) passes, the more GRRM weaves the stories together. With the first novella, all we got was a story of how Dunk & Egg met. In short, a story of one of the Targaryen Kings and his Kingsguard. We saw many other prominent names from Westeros' history, which is truly interesting and engrossing, if you are, like I am, fascinated with the world of Westeros.
With the last novella, The Mystery Knight, we are introduced to the Blackfyre rebellion, and certain events that tie directly to stories and bloodlines in ASOIAF. There are literally a thousand and one links between the stories, and even someone as thick as a castle wall can ferret them out.
If you study the books in ASOIAF, you will find many references to people, living and dead, that have appeared or been mentioned in Dunk & Egg. Most notably, IMO, a living descendant of Ser Duncan, and a living Targaryen blood relative of the line of Aegon V (minor spoiler mentioned below). And events, most notably the tragedy at Summerhall - the childhood home of King Aegon V (Egg).
It is also noted in ASOIAF, most recently in A Dance with Dragons, I believe, that according to prophecy, The Prince that was Promised would be born of the line of "mad" King Aerys II, who coincidentally is the grandson of Egg (Aegon V). The events of Summerhall, are also tied to this prophecy.
So, I would say that the stories are very much interwoven, and that we will be learning much of the background of ASOIAF in the Dunk & Egg series, and vice versa. I found the Dunk & Egg novellas to be fascinating reads, and found myself rereading both them and ASOIAF to investigate the connections.
Spoilers to follow.
Spoilers for the fate of Dunk & Egg:
Dunk & Egg both perish in the tragedy at Summerhall. Reputedly in an attempt to wake dragons. This is mentioned many times in ASOIAF, and it is also noted that prince Rhaegar Targaryen often thought with some sadness of Summerhall.
This tragedy is said to have occurred because of a certain woods witch connected to Jenny of Oldstones and the Prince of Dragonflies (who might be Aegon V's son and heir). This woods witch is believed to have appeared in an Arya chapter in ASOIAF as the ghost of high heart.
Minor spoilers for ASOIAF:
Maester Aemon (the very old maester in the service of the Night's Watch) is Egg's brother. When he joined the Night's Watch, he was escorted there by Bloodraven, Egg's bastard great-uncle. Bloodraven later became Lord Commander of the Night's Watch. More of this story will also be revealed in A Dance with Dragons.