Why is it that we cannot use species density (e.g., number of species per square meter) to
compare diversity between two sites, while it is perfectly fine to use tree density (e.g.,
number of trees per square meter) to compare density between two sites.

This may be a good place to start:
http://en.wikipedia.org/wiki/Measuremen … odiversity

and to add - density of species ie number/m2 tells you very little about their diversity.