Sex is determined by gametes and is what differentiates males from females. It is the same regardless of time, culture, social class or age.
Gender is a social construct, and can vary. It is a series of norms that are attributed to people, usually (if not always) on the basis of sex. So the idea of women being nurturing and men being providers is a gender norm. Gender norms vary across time (eg the parts of the body that can be displayed by men and women), culture (ditto, but also things like who can drive, sexual freedoms etc), culture (marriage or childbirth customs amongst many others) social class (appropriate jobs/careers) and age (hobbies, interests etc). Fashion plays a part, too (eg tattoos used to be almost exclusively male in western culture, but they are no longer gendered. Clearly these things are not set in stone, and many people cross gender norms all the time, particularly where clothing and things like drinking alcohol and fashion are concerned.
The conflict occurs when people say that they can change sex. This is a biological impossibility, so there are those who try to make gender and sex interchangeable. It is possible for people to adopt characteristics of the other gender, and most of us do so routinely, and change back and forth without insisting that we are sometimes male and sometimes female and mostly we don't see ourselves as either.
Whether people call themselves male, female or gender-neutral is unimportant, except when it allows male-bodied people to enter spaces that have been designated single-sex for reasons of safety or dignity (eg prison cells, changing rooms or rape suites). There are also issues around sport, research statistics and other areas, but they are the main bones of contention, as well as the way in which the insistence that people can change sex is mangling the language, removing words such as 'women' or 'mothers', which is, arguably, eradicating women and subsuming them into a 'not male' subgroup, and many feminists object to this.