Well, biological sex is the parts you were born with.
Gender is how you perceive yourself.
So the short and messy form is, sex=body, gender=mind. For most people these work out to be the same thing, which is why it's typically used interchangeably. For those of us who are transgender though, that isn't necessarily true.
So, would I change my gender (female)? No. Hell no. That would be destroying my identity, I couldn't stand that.
Would I change my sex (born male)? Hell yes. I am in the process of doing that now, since I am transsexual.
My greatest wish in life is to have my body match what my brain tells me it should be. I spent more than the first half of my life trying to make my brain match my body and it nearly killed me. I've spent most of the second half of my life trying to live with the difference and it nearly killed me. So the only thing I have left is to make my body match my brain, and life finally is worth living to me now that I am. I wish I had started sooner.