The guest is afraid of Dr. Livesey because the doctor does not back down.
The new guest at the Admiral Benbow is a rough man. He is actually the pirate Billy Bones. He likes to drink, sing, and tell stories. The villagers are both afraid of him and excited by him. He does not pay his bill, and the old innkeeper gets sick. When the doctor is called, the guest seems unable to bully him like the others.
[It was a] contrast the neat, bright doctor, with his powder as white as snow, and his bright, black eyes and pleasant manners, made with the coltish country folk, and above all, with that filthy, heavy, bleared scarecrow of a pirate of ours, sitting far gone in rum, with his arms on the table. (ch 1)
When the guest begins singing, again, the doctor tells him he is going to die of drinking and “you shall hang at the next assizes.” The guest backs down because he is not used to anyone standing up to him. He typically frightens everyone with his gruff appearance and violent songs.