According to this text, how is "I am well" different from "I am feeling well"?

"I am well" indicates that this person is in a healthy state and wasn't previously/currently sick or in a troubled state.

"I am feeling well" indicates this person (for an example) was recently asked if they were doing okay, meaning they were sick, troubled, etc ...

Both were responding to an individual about their current state in a difficult difference of their apparent meaning.