Comparing the two version, the first theme where the emphasis in placed is, I believe, the father figure importance. Compared to the Gaimar's version, which keeps it short and sweet with the context of relations, the other version goes above and beyond to stress that king Athelword was a fair, respectful, religious, moral, protective and thoughtful man (and father.) His death thus evokes much more emotions than the Hamar's version ever could. He also is much more pious and stresses faith much more that Gaimar's king. It goes the same for his daughter, who is claimed to be thriving for she had been raised kingly enough (65), however the other version portrays her as the typical white fairy-like lady; a beautiful woman, the fairiest alive, wise in all good and valuable manners.
Additionally, in both versions I noticed a notion of (not) having enough to consume, and thus being able to survive. In Gaimar there are the barons having tons of food that they are even able to share, while in the other version Havelok cries of hunger. This draws a particular picture of fighting for survival and the food being the basis for it. All the suffering and hunger might be there to display the chaos and terrible conditions during the conflict. Havelok was the only child left to live, which creates that image of him being chosen from the start. I did not really get that version of Havelok in Gaimar, as he sort of jut comes from the woods, no mention about his upbringing or any of that sort.