Great questions! I'm more of an numerical ecologist than a 'real' statisticien, but hopefully I can help.
Here, the unifrac distance is being 'weighted' (or not!) by the abundance of the ASVs you are comparing between samples. Unweighted compares all differences in branch lengths equally, while weighted gives more weight to ASV based on their abundance in the data set.
This means that when you look at all ASVs there's a significant difference (p=0.043), but the most common ASVs that make up most of the community are not very different (p=0.319).
It's not caused by a difference in dispersion. But yeah.
Check out these posts! (probably easier to read then a stats paper, and with more emoji )
And of course, let us know if you have any questions!