What problems are you referring to.. i dont see any banding ... the gradations look a bit different but both seem very smooth.

There is banding. The word you are looking for is aliasing. In your case it happens because a float value - your color channels values from the pixel shader - will be rounded to the target format's resolution (bit depth of the color channels). I'm judging this from your PNG-screenshot which has 24-bit per pixel, i.e. 8 bits per channel. [0..1] will be scaled to [0..255].

To remove this banding "automatically" you have to render to a format which has more than these 8 bits per channel. This has to be supported by your graphics card: Most (modern) cards can render to even a float valued target, but frame buffer (the device back buffer which will be sent to to your monitor) formats higher than 8 bits per channel is what you need in this case. Not all cards can do that (e.g. my GeForce 8500 GT can only do 8-bit and, well, less).

Then: Your monitor must cope with such a higher color resolution. Not sure, but the term is AFAIK contrast. But I don't know what connection quality (HDMI, DVI, SVGA ?) is needed to make this work.

You can "remove" banding in lower resolutions by dithering.

Edit: Forgot to tell you that blurring does not remove the banding.

This could be due to a million things, but the one that comes to mind first is gamma correction. Are you gamma correcting? Or...perhaps you are over gamma correcting? The picture you provided doesn't provide much black. Perhaps you could show one that goes from white to black?

The Gaussian image looks very smooth on my monitor. There is no noticeable banding unless you look really closely and really hard.

The RAW image has horrible banding.

Perhaps it it your monitor? Some LCD displays have 6-bit depth instead of 8. In any case, it could just as easily be your monitor or it's calibration, not the image.

Your image has all the possible values in the range. It doesn't skip even one color in a 'band' that I could find, but goes ... 253 252 251 250 249 ..., and any visible banding is either because your eyes can see a difference between those colors or because your monitor isn't good enough. I believe many cheaper monitors actually don't even have 255 levels, but fake it with patterns.
You could try to fix it with dithering. I saved the following image by converting your image to 16 bit per channel in photoshop, then blurring it and converting it back to 8 bit. It seems to have some dithering to hide the bands.

Forget to mention how one can see banding: Zoom in and out of the picture fast and repeatedly. The eye can detect change in contrast better when something moves. Also: contrast is perceived worse in the center of your focus. Try this at a clear night: Look slightly past a weakly shining star. You will see it better than if you were looking directly at it.