Thanks for the map Dave!
Its strange that the Washington County (formerly the most Republican in the state) went for Chaffee, but Providence County (still the most Democratic) was Laffey's best area.
My guess (and I'll admit I know very little about RI) is that Washington is a traditional Republican county and thus more likely to be loyal to the party establishment, and also may be more of a Rockefeller Republican type area due to its traditional GOP ties. Providence may be more polarized, have more Republicans who have moved into the state from other areas and thus may be more conservative due to it being a larger city and thus perhaps more likely to attract immigrants from out of state, and also there may have been more Democrats there voting for Laffey to enhance their party's chances of winning the Senate race (since this was an open primary).