Nice question... like Lucius said, I think there are many physical differences, especially in how they work at the cell level (structure, organelles....etc)...
I think the greatest distinction would be the case for animal rights (we can get all philosophical here, but that?s not my point). The fact that we have laws that acknowledge and protect animal rights, we have veterinarians, and their bodies somewhat resemble ours, are arguments that ?demonstrate? some similarity between humans and animals? I don?t think you can make a case for similarities between plants and humans?.
Maybe the only difference is subjective? in our heads, based on what we deem as valuable and important, like the fact that it is more important (and more common) the bond that we can have with an animal than with a plant.
Among other things they are not from the same kingdom? they are different? not one more important than the other.. just different?.
Always remember.. fed your animals and water your plants?.