The best way to cut an image out perfectly is to outline the item with the pen tool. This way you draw the outline around the object and cut it out. You then have complete control of how smooth you want the image is. The downfall of this is that it takes a lot of time to do this on complicated shapes.
To get a quicker cut out, use the magic wand and play with the tolerance values to allow more similar color values to be picked up by the wand. The default is 32, a higher number will select more colors, and a smaller number will select less.
After you select the white color, and delete it, if you see there is still that annoying white line, you can go to modify selection and pick increase selection size by 1 pixel, then hit delete. This should get ride of that outline.