I don't see a picture in your post, but my Blu-rays do not show any really noticeable grain - there is a bit, but I only really notice it if I get right up to the TV screen. By their very nature the Blu-rays should always be better quality than anything that is streaming, since streaming compresses the data, whereas the Blu-rays should be true 1080p. The versions that HBO stream are the same as what's on the Blu-ray, only that one is streamed and one isn't, I don't see why HBO would be any better.
That said, I only have the DVDs and the iTunes downloads to go by, we don't have HBO here in the UK so I can't compare to that. If the HBO version is even cleaner then it must look amazing, because I think the Blu-rays look fantastic.
I can't take a screenshot, but I did try and take a picture of my TV screen to post here but it was all blurry, so won't be much help.
There should be comparisons between the Blue-rays and other versions on the net, I remember seeing some , I'll see if I can find them.
Plus the added benefit of having the Blu-rays is that the episodes are yours forever, if HBO take them down then you've lost them.