In the grand scheme of things, Bilbo was arguably the least important member of the quest for Erebor (i.e., the quest of the Dwarves in The Hobbit). The other members of the quest certainly appreciated his contribution, but the most impressive feats accomplished by the quest were all performed by the others: Thorin reclaimed Erebor as his kingdom; Bard killed Smaug; Dain succeeded Thorin upon the death of the latter; Gandalf (unbeknownst to everyone else) dealt with the Necromancer.
Bilbo's main contributions were relatively unimportant to outside observers: he helped save the other members of the group at least once or twice. He found a ring, which was later revealed to be the One Ring, but at the time, everyone assumed that it was just a neat ring that makes the person who is wearing it invisible.
All the characters who knew how important Bilbo was to the quest were relatively unsociable: Either Dwarves who tend to stay away from non-Dwarves, or Gandalf, who has much more important and interesting things to do than to promote an unknown hobbit from the Shire.
Aside from these characters, only other hobbits would have known who Bilbo was, and they were never likely to tell everyone they met about the incredible hobbit down the road.
First of all, other hobbits didn't believe Bilbo's stories. They had always been a bit suspicious of him, due to his tendency to wander off in search of adventures. When he disappeared for over a year (something hobbits frown upon), then came back and started telling his neighbors insane stories about talking spiders, trolls being turned into stone, dragons, and wizards, they all assumed that he had finally lost his mind.
Second, hobbits don't approve of the kinds of things that Bilbo claimed to have done. They dislike adventure, they scorn people who they deem to be unreliable and unpredictable, and they distrust anyone who seems to be impressed with themselves. Bilbo was guilty of all these offenses, and seemed to be totally unrepentant. This is why, in the opening chapter of The Lord of the Rings, Tolkien emphasizes that Bilbo's party was the first time, since his so-called "adventure", that his neighbors thought of him as anything other than a shady character.
In the eyes of the outside world, Bilbo hadn't done anything worth mentioning; in the eyes of the hobbits of the Shire, everything he claimed to have done made him that much more suspicious.