I generally consider sex to be physical, and gender to be mental. And when I use the terms "male" and "female", I usually mean their sex, not their gender.
This is what bugs me about a certain breed of SJAs - they use "male" and "female" to refer to gender, so they can complain when people refer to issues exclusive to one sex as "male-only". An example would be an SJA saying that there are men who have to deal with menstruation (meaning pre-op FtM transgender people).
Given how the SJA crowd try to be over the top with regards to being politically correct their use(abuse) of the terms does not surprise me much. In general, male and female refer to ones gender. Hence any trans person will always referred to correctly and any non trans person does not care to begin with. However, this is more for general conversation, something SJA's manage to screw up on a regular basis.
More accurately Sex is the physical biological organs and attributes relating to reproduction.
We have primary and secondary sexual characteristics, where ones genitals are the primary sexual characteristics and the rest that you get during puberty are the secondary sexual characteristics. Sex as a term in general use normally refers only to the genitals of the person in question not the secondary sexual characteristics.
As a side note sex is also used as a distinction on a genetic level where the Y chromosome is considered male. However given how genetics can be messed up with multiple X's and Y's it is not a reliable indicator of biological sex.
Gender is entirely ones gender identity. It is used exclusively in this regard, but many confuse it with sex as in general society where trans-people are relatively rare they can be used interchangeably. This does have a biological basis in the brain as male and female brains are different structurally. (trans people generally have a brain structure closer to their preferred gender than their sex.) Gender is hence also static and innate given the biological link. This ties in with sexual orientation, but only in that both are decided statically in the brain.
Gender roles are entirely societal. There are some innate gender "roles" but these are more akin to reflexes that preserve the species and not the society. (these would be things like sexual partner selection, protecting ones young, etc. Nothing at all to do with generally accepted gender roles.)
The question would have to be, are these actually different things? Do the separate definitions actually mean anything? Am I safe in calling bullshit on people who claim their gender changes based on the room temperature, or will that piss people off too?
You are perfectly safe in calling bullshit on someone who claims their gender changes based on room temperature. Their perceived gender may change, but their brains do not. They probably have a gender ambiguous brain structure and switch between masculine and feminine habits (gender roles) depending on the situation, but their base gender remains the same, somewhere in the middle but probably leaning ever so slightly to one side or the other.