I think the two mains reasons behind this are:
That Sauron is slowly increasing his powers over the whole timeline of the Hobbit and the Lord of the Rings - the stories of which take place over an 80 year period.
That Bilbo never receives the attention of Sauron in quite the same way that Frodo does.
Bilbo's 111th birthday party which starts the story of the Lord of the Rings, and where Bilbo gives the ring to Frodo, takes place about 60 years after the events of The Hobbit. A further 18 years elapses before Frodo starts to bear the ring towards Mordor to destroy it.
In the distant past, Sauron was defeated by Isildur and thought to be destroyed, however he is slowly rebuilding his power for literally hundreds of years under the guise of being The Necromancer of Dol Guldur. Sauron does not even openly declare himself until about 10 years after Bilbo finds the ring, and he believes the ring has been lost in the area he was defeated by Isildur.
So at the time that Bilbo owns the ring, Sauron is not as powerful, and he believes the ring is lost and is searching for it in the wrong area.
In Tolkein's writings, Bilbo's party is in S.R. (Shire Reckoning) 1401. It is not until some time between 1409 and 1417 that Gollum (venturing into Mordor seeking the ring himself) is captured by Sauron. Since Gollum is released only a year before the Nazgul actively start seeking for "Baggins" one can guess that it is likely that this happens towards the end of this time period.
So it is well after Bilbo is no longer the owner and active user of the ring that Sauron starts to seek him. Frodo obeys Gandalf's recommendations and does not use the ring in this period, and breaks this for the first time 'accidentally' in the Prancing Pony very briefly, and does not seem to suffer particularly from the experience.
It is when he is attacked by the Nazgul on Weathertop that he first starts to see things - and even then it is primarily the nearby Nazgul that can sense and see him - not Sauron yet from his tower hundreds of miles away.