Out of all the seasons, Tolkien's use of autumn has the most resonance. Not only does it allude to the end of the journey and foreshadow possible decay and death, but Tolkien also uses Durin Day as a significant plot point in the story. The moon letters written on the Thror's map that Gandalf had received from Thain II predicted that on Durin's Day "the last light of the Sun as night fell" would reveal the secret door into the Lonely Mountain. So with the clue on the map, Tolkien also uses this significant aspect of the season as a transition for the adventure.
The seasons follow the archetypal hero's journey. I agree with what was mentioned above. Archetypically, spring represents rebirth, renewal and potential. Summer represents aging, sometimes accompanied with wisdom. Winter represents death, sadness and dormancy. Autumn is decay and leads to death. By using the seasons at the stage in the hero's journey, Tolkien can reinforce the mood with seasonal imagery.
Gandalf arrives in Spring, which symbolizes rebirth and new beginnings. For Bilbo, he is about to be reborn into his "Took" nature. During the summer, while the company is journeying, Bilbo grows in his heroism and becomes more self-reliant in solving problems instead of relying on Gandalf. It is in the fall when Bilbo faces the dragon, Smaug, and when he uses the Arkenstone as a bargaining tool in an effort to prevent war against the dwarves. This is a huge change in Bilbo (Autumn symbolizes change). Winter symbolizes death, and as the war concludes and Bilbo begins his journey back to the Shire, the old Bilbo Baggins is "dead" (he is even given up for dead by the members of the Shire). His return to the Shire one year and a day after he left (it is again Spring) symbolizes the full emergence of Bilbo's "Took" side, as he is now an adventurous Hobbit.
What an interesting question. I must admit I didn't think of this. I do think the text places more emphasis on setting and the weather rather than seasons - for example how the setting changes as the Company leaves civilised land and the weather matches this in a kind of pathetic fallacy. I guess you could say that the changing of the seasons mirrors the development of Bilbo's character in terms of signifying the complexity of life that Bilbo faces in his adventures and the challenging decisions he is forced to make.