Why do you think the overcoat made such a difference in Akaky life?
Gogol is not oblivious to the real physical needs of people and to the way his society could put meeting those needs out of reach or make obtaining them very difficult. On a practical level, therefore, Akaky's warm new overcoat helps keep him alive, at least briefly, in the freezing cold winter weather. His old coat had worn out and Akaky had scrimped and saved for months to buy his new coat.
Akaky is also treated differently when he comes into the office in his new coat. Rather than being ridiculed and bullied, his former situation, he is now praised and honored. The overcoat reveals that people judge others superficially, not on the basis of who they are inside or because they have intrinsic value as human beings, but because of what they own or how they present themselves. The outer garment--the "overcoat"-- truly does matter. This trait in humans of judging others by appearance will again work against Akaky. When Akaky's coat is stolen, he returns to being the insignificant person he was before. This makes it impossible for him to achieve justice, and on a literal, physical level, leaves him exposed to brutally cold weather so that he gets sick and dies.
Akaky is a little man with little recognition. He works hard for little money and no praise. His co-workers make fun of him and don't really pay any positive attention to him until he buys his overcoat. Akaky had to sacrifice and save for months and months to pay for that overcoat. As soon as he walked into work with the new coat on he was suddenly the center of attention. He had never gotten so much attention. His co-workers talked their supervisor into having a little party in honor of Akaky's new coat. On the way home from the party the coat is stolen and through his experiences in trying to find justice, he realizes he is back to being a man of no import.