How many ways are there to put 8 beads of different colors on the vertices of a cube, if rotations of the cube (but not reflections) are considered the same?

Actually, this is the correct answer.

Consider one vertex of the cube. When the cube is rotated, there are 8 vertices which that vertex could end up at. At each of those vertices, there are 3 ways to rotate the cube onto itself with that vertex fixed. So, there are a total of 8*3=24 ways to rotate a cube. There are 8! ways to arrange the beads, not considering rotations. Since the arrangements come in groups of 24 equivalent arrangements, the actual number of ways to arrange the beads is 8!/24=1680.

What is being requested, if I'm not mistaken, is the number of permutations for placing each of the 8 beads on the vertices of the cubes;

In this case, we have 8 different beads and 8 possible locations for each of them;

So the number of permutations is:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320