Kind of, Normal maps are more complex.

A quick google brought this up:

  • Bumpmaps:
    Single axis hieght information maps that the renderer can use to "shade"
    the surface it is applied to. (1D Bump Maps). The hieght info is usually
    stored as a grey scale 8 or 16 bits per pixel image file where the low
    values (black) represent the low areas (valleys) and the high vlaues
    (white) are used to represent the the high areas (peeks).
  • Normalmaps:
    Three axis hieght information maps that the renderer can use to "shade"
    the surface it is applied to. (3D Bump Maps). The X, Y, or Z axis specific
    hieght info is usually stored as a 8 or 16 bits per pixel color image file
    where the low values for the x axis (black) represent the low areas
    (x-axis-dents) and the high vlaues (RED) are used to represent the
    high areas (x-axis-protrusions). Green is used for the y axis in the
    same way and blue for the z axis. Because normal maps contain
    hieght information for all three axis (x, y, z) it is posible to discribe
    a complete 360 degree curvature on all 3 axis in the shaded surface.