I work for a man who used to be an Honorary Consul. The H.C. plate is issued by the state and doesn’t really carry any special privileges at all: you don’t get out of parking tickets, you can’t park in handicapped or otherwise reserved spaces, you can’t speed, etc. In my state (MA), anyway, the main advantage is that the cost of the plate is much less than a regular plate; and it is possible that cops/meter maids might let things slide for which they would ticket a regular person, but they are not under any obligation to do so.
The office of Honorary Consul is just that - an honorary position - and doesn’t carry any diplomatic benefits. Most H.C.s are business people who are granted the title in exchange for providing office space, funding, staffing, etc., for a Consulate branch office, where people can register births & marriages, apply for visas, have documents translated, etc. They are not the same type of office as the main Consulate that deals directly with the government of the home country.