Compare and contrast how modern classification systems differ from those used by Aristotle and Linnaeus?

Aristotle had recognized a  difference in life and arranged life hierarchically based on complexity(you can think about it like a ladder, each rung up a bit more complex then the one below, ex: snail lower than fish, fish lower than dog....etc....).
Including humans and deities at the top. recall that both aristotle and linnaeus believed in the fixity of species.
Linnaeus was inspired by the works of John Ray.  John Ray actually coined the word species.  He recognized species; as organisms that were able to interbreed and reproduce viable offspring.  Ray further noted similarities between organisms and classified them into a Genus.
This is essentially the binomial nomenclature that would be further developed by Linnaeus.  
So, Linnaeus takes it a bit further and introduces class and order, (sort of hierarchical) and begins categorizing the creatures.  Linnaeus names humans, using latin and greek forms for naming. Such as: Homo sapien, which means wise man.  and... well... not going to spoil anything here, but we still use this convention. (you can discern similarities)
more categories have been added to modern conventions. such as the kingdoms, the subs, phylum, ya-de-da, you can look that up in wikipedia under taxonomic systems.
Linneus, aristotle and most everybody else pre-1900 didn't understand a lick of genetics, DNA structures, theories of evolution, which can account for ancestry and demonstrate more relationships.  I do not think that I have adequately answered your question; but I do feel it is a good spring board for you to start upon.
The similarity is that there is a recognition of diversity.
The difference is the understanding of the underlying similarity.