In a 256x256-pixel image you’re working with 65,536 pixels. With 1068x1068, you’re working with 1,140,624 pixels, which is what, over 17 times more pixels. In other words, you’ll never get the detail in the smaller image with fewer pixels that you got in the larger one with more pixels. However, since the sprites will, I assume, be viewed on a display, that large image will appear exactly the same as the small image when the allocated space in the browser or app or whatever is specified to be 256x256 pixels.
All that considered, I don’t really know what you mean by the quality deteriorating with the smaller image. It could be just the result of fewer pixels, like I mentioned above, or it could be the result of some other thing that’s happened that has just messed up the image somehow, like jpeg compression or something along those lines.
What you likely could do, if you like the way the larger image looks, is to just sample it down to 256x266 pixels in Photoshop. In other words reduce the size of the larger image to that of the smaller image in Photoshop. However that ends up looking, is probably the best you’re going to get.