So, what does this say about what the story is? Why did Dickens write it? There is strong indication that Dickens saw parallels between 1775 and his own time, particularly when it comes to the subject of revolution and "mob rule", as pointed out in the lecture previous to this post. These contrasting elements serve multiple purposes. On one level, it allows Dickens to easily relate a historical story to modern readers. On another, it provides him with a structure through which he can highlight specific aspects of society (the disparity between the wealthy and the poor is a major one).
A particularly striking example is in chapter five, "The Wine-Shop". The chapter opens with a cask of wine breaking open in the streets. For a few precious moments, the crowd is overcome in manic joy. However, once all the wine is lapped up and only mud is left, the citizens retreat back to their dreary, depressing lives. As far as the core of the plot goes, this adds very little. Its sole purpose is the establish an environment in which both extreme joy and sadness can exist closely together.
Further support comes in the form of Lucie's relationship to her father. In many ways, Lucie is a foil for Dr. Manette, at least upon their introduction. While Lucie is described as an embodiment of youthful beauty, her father is a decrepit shell. One is full of life, the other drained of it. And yet, despite their differences, they are brought to an emotional level upon meeting each other -- there is much sobbing and confusion to be had.