The reason why one should "Judge the sin, not the sinner". We cannot judge other people for the following reasons:

* We don't have the right (permission)
* We don't have the responsibility (it is not our job, nor is it expected from us)
* We don't have the power (to enforce judgment of a person in a correct manner, a judgment without enforcing is meaningless)
* We don't have the ability (to judge a person by pure standards, to weigh of the total of good and bad acts a person has done)
* We don't have the insight (we can't be certain about what the intention, background of a person was)

The reason why not to hate the sinner, and only the sin:
First of all; one shouldn't hate something/someone when it is not for the sake of Allah (subhana wa ta'ala). Secondly, we know that Allah (subhana wa ta'ala) hates sin, but he doesn't hate every person that has committed a sin (if not he would hate everybody). And since we cannot judge a person, we don't know who is hated by Allah (subhana wa ta'ala). In fact in many hadeeth the prophet (peace be upon him) told us about people that were considered sinners by society, but in the end will be forgiven by Allah (subhana wa ta'ala). So when you hate a sinner, it could be that you hate someone who is loved by Allah subhana wa ta'ala.

Ps: this is actually the same thing as you'll find in western law. As example, a person who is on trial for racism cannot be convicted for being a racist (he cannot be convicted merely on the basis that the jury indeed establishes that he holds racial views and opinions), but he can be convicted of committing racism or spreading racism. So he is on trial for his actions, not for who he is.