Why do sociologists find it important to differentiate between sex and gender? What importance does the differentiation have in modern society?

I would add to the excellent points made by the first educator that the traditional understanding of "sex" did not allow for the fact that a significant minority of people—about 1% of the global population—are intersex. This means that they do not have the genitalia and reproductive equipment we identify as being either "male" or "female." Historically, it was common to try to turn all intersex babies into girls, but a good understanding of what gender is has allowed for advances in the way intersex people are treated. Rather than attempting to force intersex babies to identify as female, studies have found that the natural gender traits of intersex people will often exert themselves, regardless of how they were socialized as children, such that a child who has been raised as a girl may experience gender dysphoria as a teenager upon recognizing that he feels himself to be a boy. As such, there is now a strong argument for raising intersex babies without specific gender until they hit puberty and allowing these children, who are not defined by any specific binary "sex", to determine their own gender.
Conflating "sex" with "gender," then, is problematic because although many people Assigned Female At Birth (female-sexed) do grow up to feel that their gender expression is female, this is not always the case because sex is a physical trait, and gender is, indeed, a matter of self-expression. This idea becomes more obvious when we consider those people who do not have a defined sex; historically this caused panicked attempts to carve out a "gender" box for them, whereas in fact, we are now learning that our gender is something we can define for ourselves and need not correspond with any defined biological sex.

In sociology, sex refers to the biological characteristics that a society uses to categorize people as either male or female. For example, each of the sexes has specific genitalia. Other secondary characteristics that are acquired upon adolescence are the development of breasts and broadening of hips among girls and the deepening of voices among boys.
Gender, on the other hand, explains how societies identify and take care of the different sexes. This includes a cultural interpretation of the roles played by either sex in the society. It also looks at how the society influences our understanding of the sexes. It addresses what is seen as "male" or "female" behavior—aspects that are not defined by the male or female biology. It addresses questions such as "can a man wear a dress?" or "why do we have distinct names that separate boys from girls?”
The distinction between sex and gender is important, because sometimes a person’s sex may not align with his or her gender. There is pretty clear evidence that gender is socially constructed. Different societies can ascribe different behavior to a gender. For instance, while it is weird for men to wear dresses in certain societies, other societies in Africa, Asia, and the Middle East have a culture that encourages the wearing of dresses or even skirts among men.
In modern society, a distinction between sex and gender is also important because of changing gender roles, for instance in occupations. Women are increasingly taking up jobs that were previously considered "male jobs," (e.g., heavy truck driving, technology jobs or even politics). As such, the term "gender" rather than "sex" is used to remove traditional societal biases between the sexes.
https://www.monash.edu/medicine