The same image 1024x1204, PNG format, resized to various sizes on https://pgicons.abiro.com will look great. But, if I resize that same image directly in Photoshop, let's say to 96x96, it will not look as great.
96x96 generated automatically on pgicons.abiro.com:
96x96 in Photoshop CS6:
I don't understand why I can't achieve that same quality in PS.
PNG is a compressed image format. I'm afraid my monitor's not showing me a difference between your shared images, but I assume that since the website you mention specializes in ONLY resizing PNG's, it has a particular algorithm to do so.
You're using a quite old version of Photoshop. Not a problem, but likely not as updated for working with compressed PNG's. Maybe resizing from your original image will provide a better result out of Photoshop.