The most significant thing that is implied or meant by the (to us) unusual punctuation is that conventions and rules of punctuation--like rules of spelling--have changed over the millennia. Also like spelling, there were periods during which there were no rules, or only vague rules, and merely conventions of punctuation. To illustrate this idea, one of the great complaints in early periods in regard to punctuation was against women letter writers who were said to use no end stops in their letters (end stop: period, question mark or exclamation mark). This was a convention of punctuation that seemed gender related (it was actually related to style of education).
Recalling that rules of punctuation were not as firmly fixed as they are now, Dickens employed or broke rules of punctuation--or employed or broke conventions of punctuation--just as Shakespeare earlier broke rules or conventions of language. Shakespeare manipulated the rules of language to fit the expression of his intentions; he didn't fit the expression of his intentions to the rules of language. Dickens uses or breaks the rules or conventions of punctuation to fit the expression of his intention; he doesn't fit the expression of his intention to the form of punctuation rules or conventions. As a side note, an America poet who similarly ignored rules and conventions of punctuation was Emily Dickinson.
The reason Dickens chose the punctuation convention he employed in the opening paragraph is to show each item as equal to its antithesis, and each set equal to the other sets. If he had used semicolons, there would have been a distinguishing of sets of ideas and this distinction would have overridden any sense of equality between sets of antithetical ideas. What Dickens achieves with the commas is equality within antithetical (or binary) pairs and equality between antithetical (or binary) sets. Only this leveling, this wholesale equality of ideas gives his intended picture, his intended image, of the era: it was an era of equalizing.
In addition, this leveling and equalizing of importance and weight allows Dickens to slip into an ironic description and satirize his own era as one that is exactly like the year 1775 and as one that can be described with the "superlative degree of comparison" only. Superlatives, you may recall, are adjectives and adverbs that indicate the best or worst and the most or least of something. Thus, while Dickens says his own age is one that can be described "for good or for evil, in the superlative degree of comparison only," you will note that Dickens also describes 1775 in the superlative degree of comparison and that he provides his own explanation--thus embedded in the text--for doing so: to describe it adequately requires the superlative degree of comparison only.
IT WAS the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we had nothing before us, we were all going direct to Heaven, we were all going direct the other way- in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, in the superlative degree of comparison only.