The character played by Don Knotts, apparently "Barney Fife" from the old show, is never addressed or referred to by any name, even though he calls Andy by the nickname "Ange" just like Barney did in the first Andy Griffith Show. The closest anyone comes is when Andy's young son just says he's Andy's "old friend".
The plot in this episode is similar to that of The Andy Griffith Show season 8 episode 27 "Sam For Town Council" with Sam Jones being pestered by people seeking favors now that he's been elected to Mayberry town council. Here Andy Sawyer faces the same predicament being the new mayor of Greenwood.
This is Barney's, Emmett's and Goober's only appearance in the series. They all claim to be old friends of Andy's. But why he changed his name to Sawyer, and what's become of Helen and Opie, no-one knows.