Centriolesare useful, but not extremely essential in
cell division, but
spindle fibersare. It is the spindle fibers that pull the sister chromatids apart during prophase in mitosis, and not the centrioles. The centrioles merely help in the orientation or organization of the spindle fibers (which are microtubules). This means, the centrioles determine where exactly in the cell the spindle fiber develops.
However, the spindle fiber can be formed even without the centrioles.
I do not why the animal cell requires organization in the assembly of spindle fibers, whereas the plant cell doesn't, but I guess it is because cytokinesis (division) of plant cells occurs by cell plate formation, whereas in an animal cell, cytokinesis occurs by means of furrows or cleavages.
Because of the (more or less) regular shapes of most plant cells (because of their cell walls), the cell plate will always be formed in the same position regardless of its orientation. However, in an animal cell, which is highly irregular, the spindle fibers may develop across any part of the cell, thus cleavages may form at any part of the cell. Hence, to ensure that the cleavages develop exactly at the equator of the cell, the centrioles are important.
