Why doesn't Bob recognize Jimmy Wells when he sees him in "After Twenty Years" by O. Henry?

This is a valid question; O. Henry must have thought about it himself. O. Henry wanted Jimmy to stop in front of Bob without Bob recognizing him, so he invented several reasons to explain why Bob would not do so.

For one thing, when we are meeting someone we haven't seen in twenty years, it is easy not to recognize that person because we expect to see the same person we last saw twenty years...

This is a valid question; O. Henry must have thought about it himself. O. Henry wanted Jimmy to stop in front of Bob without Bob recognizing him, so he invented several reasons to explain why Bob would not do so.

For one thing, when we are meeting someone we haven't seen in twenty years, it is easy not to recognize that person because we expect to see the same person we last saw twenty years ago. One of O. Henry's points in the story is that people change a lot in twenty years. In fact, everything changes. The whole neighborhood has changed, and Bob is standing in the doorway of a closed hardware store when he expected to be standing in front of a busy restaurant. Bob and Jimmy were both very young when they parted at 'Big Joe' Brady's restaurant. Now, both are around forty. People can change a lot in appearance and manners between youth and middle-age. Bob was eighteen and Jimmy was twenty when they parted.

Any uniform tends to disguise the person wearing it. Jimmy just looks like a policeman. O. Henry specifically states,

the officer, with his stalwart form and slight swagger, made a fine picture of a guardian of the peace.

This is another way of saying Jimmy looked just like any other policeman in New York City.

O. Henry emphasizes that the whole neighborhood is dark because it is made up of little shops which are all closed by ten p.m. Bob is standing inside a darkened doorway. When Bob lights his cigar, it doesn't help him see Jimmy. The matchlight would be right between him and Jimmy and would temporarily blind him. It takes a long time to light a cigar. When the match went out, Bob would remain blinded. Anyway, Bob already believes he is talking to a cop he does not know, and Jimmy has not tried to tell him otherwise because he realized Bob is wanted by the Chicago police. As he writes in the note the plainclothes detective hands to Bob after making the arrest:

Bob: I was at the appointed place on time. When you struck the match to light your cigar I saw it was the face of the man wanted in Chicago. 

Jimmy has changed over the years. He has aged, gained weight, and taken on the mannerisms of a typical patrolman. He is disguised by his uniform and its big cap. How would Bob recognize Jimmy Wells after twenty years? Jimmy is good at his job. He doesn't talk much; he just looks and listens. Bob seems more concerned about his own appearance than the man to whom he is talking. Bob's shows his great ego through his talking and the flashy possessions he displays—a diamond scarf pin and "a handsome watch, the lids of it set with small diamonds." Bob is also nervous because he knows he is wanted in Chicago. Here he is, a wanted man who finds himself talking to a policeman! He is fooled by both Jimmy and the cop in plainclothes.