If you want to study gender from a sociological perspective, then the first point is that gender is also somewhat socially constructed. In other words, we might believe that there are only two genders - male and female, but when we begin to examine other cultures, things get more complex for two reasons. First, how do we classify hermaphodites? Some culture see them as unnatural and dangerous; other cultures see them as powerful. The point is that the concept of gender is somewhat fluid. Second, even culture that has two genders have different notions of roles among the genders. Differences will abound.
The word "gender" has at least two common meanings. The two meanings, interestingly enough, are conflicting.
The first meaning is biological sex. Often we you fill out a form, you'll see under "Gender" two boxes: female or male. "Sex" is used on forms, too, but "Gender" seems to be just as common. I wonder if people use "Gender" here as a sort of euphemism, as "Sex" has other meanings, too.
The second meaning deals not with biological sex but with the full range of social roles and performances that are commonly associated with women or men. Recent theorists (e.g. Judith Butler) have built impressive and elaborate theories around the idea of gender not as an essence (i.e. something that you are) but as a performance (i.e. something that you do).
One of the initial drives to separate the categories of "sex" and "gender" (in the sense of its second meaning) came from second-wave feminists in the late 1960s and early 1970s. They sought, in their various ways, to resist simple stereotypes about how women should behave or think. More recently, with more women in leadership positions and greater public awareness of transsexualism, etc., I think a number of people often have a fairly sophisticated sense of how biological sex and gender performance don't always match up in expected ways.