Oh, I see the confusion here. People don't usually admit to not knowing the difference between sex and gender...! 
That's a little embarrassing. Okay.
Biological sex is based on whether you have chromosomes that let you make reproductive gametes - i.e. eggs or sperm. We call animals that get pregnant or lay eggs "female" animals. We call the sperm donors "male" animals. Sex is usually determined by chromosomes. Now, as a biologist I won't confuse you with the various exceptions and details to this, even though there are lots of them :D Just remember that sex = the physical part. Chromosomes, penises, whatever - the ability to get together and reproduce!
Gender is what you "are." It's a piece of your identity. And your identity is not entirely defined by your body. If you are a man and you had your penis, testicles and prostate removed due to cancer, you'd still be a man even if you couldn't function as a member of the male sex and had no male hormones or sexual urges - your gender would still be male. If you are a woman and get a heart transplant from someone of a different sex, it becomes a female heart (gender) because it's your heart, even if it is male tissue that came from a male body (sex). You don't become 5% male because you had a blood transfusion from a male donor, either! Women with PCOS are still female even though they have lots of male hormones in their bodies. And if you had a little son and he lost his genitals in a botched circumcision then you probably wouldn't raise him as a girl, because you would recognise that he still has the ability to be "male" without a penis.
Gender doesn't always align with biological sex, and when it doesn't, that can be very sad and challenging. For example, someone with androgen insensitivity syndrome is "biologically male" (male chromosomes and internal male reproductive organs) but may never know this because their external appearance is female and they often identify as female. So these people live socially as woman, identify as women and frequently only find out about their male sex when they don't have regular periods, or have trouble conceiving. They are still female if their gender is female, though.
People with intersex genitals - who don't have an obvious sex to point to - may identify pretty firmly with a gender. Even if they have no biological sex, they can still be male or female.
Even if there was some kind of mad situation where a woman's head was put on a man's body, her gender would still be female because gender is separate from the body. Likewise, a trans woman is a woman (female gender) even if her physical body might suggest otherwise.
So that's where a lot of confusion and hurt feelings happen. People often think sex and gender are the same thing. They aren't, they're two different words with completely different biological definitions - and it's very easy to look them up if you get confused.
People are so weird, we even give gender to things that don't have genitals. Most European languages gender objects, even though objects don't have sex. In Spanish and French the moon is a girl - but in German the moon is a boy. The moon obviously doesn't have any biological sex (it doesn't have a vagina/testicles) but it has gender depending on what language you speak.
Gender is what we assign, sex is how it reproduces. If you're looking at a kitten's genitals to decide what to name it, you're sexing the kitten. If you compliment someone on their "pretty baby girl," basing the compliment on the baby's pink outfit, you're gendering the baby.
If you don't "believe" there is a difference between sex and gender, that is widely considered transphobic. So if you mix up the two, you will probably be accused of this (or you'll just sound like someone who never took a biology class... or learned another language... or had any curiosity about your body or society.) They are literally different things, and it sounds like you're saying you don't believe there is a difference between planets and stars. Or tomatoes and potatoes. Once you know the difference it isn't complicated, and you have no excuse to still sound ignorant.
And finally, gender role is what you DO with your gender. Sure you're a woman... but so what? What does that mean in your society? Gender roles change with time and place and can sometimes be oppressive. For many previous generations, women were expected to nurture children, care for the home and run domestic stuff, while the men provided all the money, because those were their clear defined roles. This has changed and now everyone is expected to do some formerly gendered stuff like cooking, cleaning and earning money. Back in the old days, typing was feminine (women learned to type so they could become secretaries) but now that everyone uses computers, typing is genderless. Pink used to be a boy color, while blue was feminine because it is associated with the Virgin Mary, but now pink is a girl color and boys won't touch it because "it's for girls."
Healing and medicine used to be a female gender role, but then the rise of modern medicine made it more masculine and women weren't allowed to do it at all - and now it's coming back around to being a feminine profession. Math used to be masculine, but now girls are encouraged to be good at math. Gender roles have expanded a lot, although there are still some things men "can't" do and vice versa. Even now, people have to justify liking/doing certain stuff and feel embarrassed for stepping outside their gender role, like my husband is embarrassed about using my spot zapper and won't buy his own, but he still secretly uses mine. Hopefully some of it will continue to change, I for one would like to see grooming return as a symbol of true masculinity, so that it is once again truly MANLY to really care about your appearance 
In some cultures gender roles are very strict and restrictive, for example, in some cultures women are expected to be very submissive and subordinate, and are punished for stepping outside their strict feminine role. In some cultures, like Italy, men can kiss and hug each other as greetings in public. In the UK, this is not considered masculine and most men would find this inappropriate/uncomfortable.
And even though gender roles seem pointless or illogical, they can be very important to people.
If a cancer patient loses her breasts/hair/eyelashes/ovaries/hormones, and her ability to care for people/cook/nurse her children, and if she doesn't look/feel pretty in the mirror, then she might feel really sad even though those are NOT the things that make up her female gender. They were key parts of her feminine identity, but losing her sex characteristics and the way she expresses her femininity doesn't make her genderless/nonbinary/a man. It still can hurt to lose them, but it doesn't mean she has lost her womanhood. Her gender doesn't change, it was always separate from her body and her role.
If a man doesn't want to use shampoo from a girly pink bottle, it's not because it's biologically incompatible with his sex, or because he thinks it will make his penis fall off, it's because his masculine gender role is strict and the pink soap isn't compatible. A man who happily uses the pink shampoo isn't more womanly, it's just that he's got a different definition of his role in masculinity and he feels it can include pink soap. The joke is "Being comfortable with my masculinity" or "being in touch with my inner woman" and men who joke about this feel that their maleness (gender) is ultimately unaffected by using women's stuff (gender role).
So in conclusion, sex, gender and gender roles are different things. And now we have no more excuses for confusion!