does order of genes in a chromosome matter?
is the difference between humans and mice due primarily to the
  different order of the genes or to the slight differences in the genes
  themselves?


The latter.


  This question was raised from a claim here which seems to imply it
  does not matter.


You are talking about two different things.  Gene expression is often controlled by nearby DNA sequences that don't code for protein.  Moving a gene away from the DNA that controls it can change its expression a lot, and have huge consequences on the organism.  But in many cases, you can move a whole gene, including its controlling sequences, to another place, and nothing much will happen.

People were bringing up synteny as evidence of evolutionary relatedness, not as the cause for all primates looking similar.