DXT & PVRTC should produce pretty different results. First thing to do is open up the sprite collection atlas texture in Photoshop and look and make sure there isn't any alpha change / black dot on the inside. Its unlikely but its worth looking at just to make sure.
I take it the sprite in your image is just the green button and doesn't include the background? If so, my prime suspect would be the background outside the white outline. Are you using premultiplied alpha by any chance?
Now, with PVRTC / DXT - Unity doesn't let you choose your compressor, but you can use your own compressed textures if you want. What I'm saying is you could load the png into ATI Compressenator / NVDXT for DXTC and PVRTEXTOOL to save out an appropriate compressed file. With these tools you can fiddle with compression parameters to try and get a bit better quality out of it. Save them out as a .pvr for PVRTC (dds for DXT) into your Unity project and override the material with it.
This is a manual process and you'd have to do it every time the sprite collection changed, but its a viable option nonetheless.
ps - I can explain why DXT looks so horrible sometimes. It works in 4x4 blocks of pixels, evenly divided from the start of your texture. (0, 0 = first block, 0, 4 = second block and so on). Each 4x4 block gets 2 16 bit colors (565) and 2 colors interpolated between these 2 colors. Each pixel in the 4x4 block can be one of these 4 colors. As you can imagine this really limits what kind of colors you can have in a 4x4 block. A sure fire way to break DXT horribly is to have a 4x4 block with different primary colors in there.