How does mitosis differ between the onion root and the roundworm?

Well, one big difference is that when the we hit telophase we start to see a cleavage furrow in animal cells but we start to see a cell plate forming in plant cells.  

Take a look at this picture of plant cells: http://student.ccbcmd.edu/~gkaiser/biotutorials/dna/mitosis/images/telophase2_pc.jpg

Now take a look at the cleavage furrow in this animal cell:
http://www.uic.edu/classes/bios/bios100/lectf03am/cleavage.jpg

What is going on is pretty different between the two.  In the case of the animal cell they split apart and in the case of the plant cell they remain in contact.

This is just one difference, but you can find more when you think about the differences in general between a plant and an animal cell.

The question is of the differences between the mitosis of a plant cell (onion root) and animal cell (roundworm).

For the most part, the phases are similar until the final phase, telophase, and a separate process called cytokinesis. I will skip the phases in which there are no differences, I assume you are aware of all them.

I. Prophase:
-- Animals: At each polar end of the cell, spindles begin to form between the chromosomes, originating from the centrosome (an area of the cell composed of two perpendicular organelles called centrioles, responsible for creating microtubules)
-- Plants: Plant cells do not have centrioles, but they do have spindles originating from their own centrosome (again, a specific area designating for the production of microtubules). The spindles form around the nuclear envelope as well as the chromosomes.

II. Telophase:
-- Animals: A contractile ring of actin & myosin forms in the middle between the two nuclei.
-- Plants: Phragmoplast (composed of actin, myosin, and microtubules) form the center of the cell where the cell wall is to be located.

III. Cytokinesis (division)
-- Animals: The contractile ring pinches the cell forming a cleavage furrow that eventually divides both daughter cells apart
-- Plants: Phragmoplast completes its formation of the cell wall and separates the two daughter cells.