My understanding is that they both have red plates, the difference is in how you engage them. TO catch a route taxi, you go to the road and wave a red plate down, giving them 100j per person (they go beach - roundabout/100j and roundabout/cliff 100j, then vice-versa). This means the taxi will stop and pick up others or may have others in the car, it can sometimes get a little tight.

A private taxi fare is negotiated with a red plate taxi driver before you get into the taxi. You can flag one down or there may be one available at your hotel. Always agree to the price before you get into the taxi. I'm not sure what the current "standard" fares are - 500j for beach/beach, cliff/cliff, then 1000j between cliff & beach? Anyway, if you do not agree on the price they may charge you more than you are expecting.