Importance of Clothes

Clothes seem to be of huge importance in Russia in this period, almost certainly due to the constant cold climate. 

The degree to which you are clad seems to refer to your social rank, so a better overcoat means that you are higher in the Russian table of ranks.

Akaky is a titular councillor, this is the 9th of 14 ranks; quite low down in the table. As a result of this, his overcoat is tattered. hardly even considered an overcoat. You can see how the instant jump in social standing from a better overcoat would appeal.