After my reading, I noticed some differences in the construction of the story. However, I must say that I am a little bit confused between the versions, so I hope that I won’t make mistakes.
For me, Gaimar's L'estoire des Engleis (modern English translation), is more focused on a women perspective. Quickly, the story settles around Argentille. She is the first to notice that the mouth of her husband is burning. At least in the beginning of the story, she seems to be a character as important as Cuaran. I also notice that Cuaran’s personality is more clearly introduced, and directly at the beginning of the story. It also struck me that the narrator emphasizes the fact that he is stupid. Even if he seems kind, he is absolutely not able to give a noble meaning to his dream. He also does not know how to act (“Argentille was in great perplexity about why he lay facedown”), and seems a bit wild as he is used to ties up his friends. The narrator seems almost sorry that Argentille is the wife of someone stupid “whatever he said, all of it was wrong!”, with a lot of exclamations.
The story of Havelok (the middle English romance) version seems quite different. Firstly, the names of the characters are not the same. Secondly, the events are not really told in the same order, from the same point of vue, or with the same intensity. I would say that the religious aspect is more present. Quite often, and already at the beginning, Christ is mentioned, which sentences such as “Let us praise the Lord!”. For me, this version is also more masculine and maybe more focused on “noble/knightly topics”, such as kingdom, heir… It is more a story of kings. The scenes are not really told from Goldeburu’s point of vue.
About the question of the form of the story, we can also notice the fact that in Gaimar's L'estoire des Engleis, the story starts directly. However, in the middle English romance, there is this kind of introduction/conclusion which allows the narrator to talk to the readers. It presents the story that we will hear, so it makes the reader want to read it, and by the way, it prays to God.