I would say the main difference here is that eukaryotic cells (e.g. animal cells) have a nucleus and membrane-bound organelles, whereas prokaryotic cells (bacteria and archaea) have no nucleus and no membrane bound organelles. Note that this doesn't mean one is more "evolved" than the other - it simply means they have evolved in different ways.
Other ways in which they are different includes size (eukaryotic cells are big, prokaryotic cells tend to be much smaller), organisation of DNA, size of ribosomes, cytoskeleton, means of reproduction and presence of metabolic pathways. All of these differences are in relation to the environment or organism they're found in, and are highly specialised.