To me, an advisor does not necessarily look out for their mentee's best interest and does not necessarily show empathy to their mentee. A mentee will get the work done with an advisor's help, but that student will do it with more happiness, pride, and satisfaction with a mentor's help.
A mentor is an advisor who goes beyond the standard description of an advisor. A mentor listens, encourages, and facilitates the opening of opportunities to come to the mentee in order to help their mentee succeed. A mentor would not hesitate to acknowledge when their student's work is being done well but an advisor may be too busy to even notice or care.
On this note, why is there such a culture of advisors being cold, absent, and too busy to even care about the wellbeing of their mentees? I feel like it is so prevalent in the graduate school community that most people accept it as fact without trying to change the situation. Why is it ok for grad students to feel like their advisor doesn't care, is too busy, and wouldn't even notice if that student were suffering or struggling? Being thrown into the deep end in order to learn how to get the job done independently is a valuable lesson to be learned and it should be the case if you want to deserve the PhD you are working for. But, aren't there simple ways where an advisor can be a true mentor?
That's my $0.02!
At first consideration, I would say the difference has to do with the level of investment. An advisor's role is to be a guide, a teacher, who can help the advisee on a path which the advisor has some experience with. A mentor, to me, is someone who is actively involved in walking the path beside the mentee and gives them necessary proddings when it's the mentee's time to stand up and shine on their own. For instance, in Tolkein's "Lord of the Rings Trilogy", Elrond was an advisor for Frodo while Gandalf was a mentor (loosely, but a fun comparison).
I vibed with both Svet's and Graham's responses :) Going along what they were talking about, I feel that a mentoring relationship is similar to an in depth apprenticeship... working closely with someone to gain a skill or meet a goal, where the mentor has just as much to gain as the mentee. Whereas an advisor simply advises on the student/individual on a path to meet the requirements. To be quite frank, my college advisor didn't give a crap about me...and I had to seek the advice of other teachers to figure our how to navigate undergrad and prepare for my future.