A double solenoid is one with two coils. Most examples of double solenoids I could find were used in solenoid valves, such as the DFRobot FIT0617 (Digi-Key 1738-FIT0617-ND) shown on the following link: https://www.digikey.com/product-detail/en/dfrobot/FIT0617/1738-FIT0617-ND/10230100
Coil A, when energized, moves the valve to position A, and coil B, when energized, moves the valve to position B. When neither coil is energized, the valve remains in its current state. This would be similar in function to a dual coil latching relay. One of the advantages is that the output controlling this valve would not have to continually stay active. A disadvantage would be that a separate output would be required to return the valve to the other position.
In contrast, a single (normal) solenoid, when de-energized, will return to its starting state, like a non-latching relay. Some advantages of this would be that only one output would be needed to control the valve, and that, whenever the program starts, it would be in a predictable state. A disadvantage would be that, to keep it in its energized state, the output controlling it would have to remain activated for the entire duration the valve is to remain in the energized state.