This discussion may contain spoilers!
WARNING MAJOR SPOILERS FOR EPISODE 4
So, there's tons of threads asking why Lee didn't amputate his hand.
There are many reason/possibility why he didn't cut it off, examples:
Bite/Scratch was minor, while Hershel's was really serious and he was losing lots of blood
He was in a hurry and cutting off hand meant less time to rescue Clementine, while Rick's main goal was to clear the prison out and make it a long-term stay home for Lori to have her baby
No medical attention/supplies (Carol, Beth, and Maggie had some medical knowledge/supplies, and Hershel is the only proper 'doctor' so he was really important to the group)