The postmaster is an educated, relatively cultured, cosmopolitan individual while Ratan is an illiterate, uncultured, orphan. Although the postmaster considers writing about the joy of nature while running the post office, the narrator informs us that he prefers a paved road to this more rural assignment in Ulapur. There is a correlation between "advanced," financial stability, and urban culture. There is also a correlation between rural life, poverty, and being less advanced.
Therefore, the postmaster represents "advanced" society and Ratan represents something comparatively more primitive. He is able to move to different places and she is not. He is in a higher social class. That, in and of itself, does not mean that higher classes tend to be more progressive and/or of the "advanced" strata of society. But, in this case, that connection is being made: that he is able to advance because of his social standing and she is not. He has had access to education and she has not. So, one comment in this story is that being "advanced" implies, first, the ability to advance. Ratan may want to advance or be "advanced" but she lacks the resources. Recall that until the postmaster offers to teach her, she did not even know how to read.
I finished this story with two senses of "advanced." The first is the ability to advance in society and to literally have the freedom to move to another place. This ability has a precursor: having some financial means. The second notion of "advanced" relates to progress, culture, and sophistication.
I think it is eventually clear that Tagore is saying that there are benefits to being advanced: being able to read, move about, make money, and have opportunities. It is even more clear that being culturally advanced (in this second sense) does not necessarily make a person more wise, more ethical, or more compassionate. If this was the case, the allegedly more advanced postmaster would have taken Ratan with him or offered her some sort alternative to start a better life. Instead, he literally and figuratively goes with the current: literally with the river, figuratively with the current of his "advanced" society. He simply moves on. Ratan can not. The final paragraph speaks to this.
In a larger context, there are many so called "advanced" societies that still engage in unethical practices, war, and so on. Being "advanced," culturally speaking, does not automatically imply that one is also enlightened.