The reason why Billy did not go to the pub is because he was already intrigued by the landlady's boarding house. In fact, he had already seen the seemingly-bewitching "BED AND BREAKFAST" sign one time. The fact that the sign kept getting his attention, and acted as a luring tool to get him inside, means that the first exposure to it would have been the first attempt at getting him to look through the window.
He stopped walking. He moved a bit closer.
When he peeked through the window that led to the inside of the boarding house, he was impressed by the chrysanthemums that were beside the green curtains that hung down on "either side of the window." He was also very impressed by the scene of comfort in front of him, complete with the fire burning in the hearth, and a dachshund that was curled up asleep by the fire. It was a tempting scene. To Billy, animals were "a good sign", meaning that the place would be quite decent to be in. He also saw the large parrot in the cage, which made the place look even more friendly.
Certainly it would be more comfortable than The Bell and Dragon.
Then Billy thought that a pub, such as The Bell and Dragon, would be more fun with beer and lots of people to talk to in the evenings. The pub would have been cheaper as well. But, just as he was about to turn and walk on to The Bell and Dragon to make his final decision,
a queer thing happened to him.
The sign caught his attention again. Over and over, he kept staring at the letters on the sign "like black eyes." Hence, he was already being hypnotized by that sign, which the landlady herself acknowledges as having the power to lure in people that she intends to get inside.
Conclusively, Billy was not able to get back to the pub because the sign had already attracted him to the landlady's bed and breakfast. That, and the other luring things that he saw inside, completely took his attention and drove him right inside.