The main visible difference between embroidery VS Cross stitch is that cross-stitch forms with counted stitches are only made with X shape stitches. It's only made on woven fabric with raised box surfaces on which you can make your design easily. You can make a cross-stitch on each box through counting the boxes and following the coloring chart on the pattern.
While embroidery is different from cross stitch because it has many variations in stitches, you don't need any specific cloth for it. You can make your desired pattern on any fabric and use colors.
Another thing that helps you understand the difference is that the work we do with the help of a needle and thread on different fabrics is called embroidery.
Another side, cross stitch is a specific form of embroidery stitch which you can make with threads. Most people use cross stitch for making borders for their craft pieces.