Huh. It’s not just that it’s pixelly-- it looks like it’s bumping areas that shouldn’t be bumped.
I wonder if some kind of dithering in the image editor you used for the blur could be responsible.
The first thing I’d do is try to use the full range for the bump map-- just auto-levels in most image editing software should take care of that. That would make any artifacts in the image more apparent.
Then I’d be making a dummy material using it as an emission shader to look for any issues with uv, uv interpolation (sphere’s a hard thing to map). And of course just to rule out everything that’s not the bump map.
If you’d like, I just made an in-material node for a 5x5 convolve (incl. gaussian blur) at https://www.blendswap.com/blends/view/90979 . You could use that to run a gaussian or box blur inside the material instead of on the image and see how it affected things.