What is the difference between dual ultrasonic sensors (sensors with two round elements) and single ultrasonic sensors (the ones with a single cylindrical sensing element)? What are the advantages/drawbacks of each design?
Sensors with two elements have separate transducers acting as the transmitter and the receiver.
A sensor with a single element uses the same transducer as a transceiver to create the sound pulse and to listen for the echo. These sensors need time to switch from one mode to the other (Tx or Rx). This means that the minimum sensing distance is likely to be longer that would be the case if separate elements were used for the transmitter and receiver.
To give a specific example, the range of the AJ-SR04M sensor is 20cm - 500cm in air. It simply cannot detect objects closer than 20cm. (The speed-of-sound in water is about 5x the speed of sound in air. If this 'waterproof' sensor is actually used underwater, the minimum distance will therefore be about 1m)
However, a benefit of sensors that employ a single element is that they are likely to require less space than a sensor with two elements. They may also be more cost effective (although this will depend on a range of other factors).
Beyond that, the relative advantages/drawbacks will depend on the particular sensor characteristics and the application(s) in which it is to be used.
Sensors Online published a two-part article titled Choosing an Ultrasonic Sensor for Proximity or Distance Measurement a couple of years ago, which was intended to help people select ultrasonic sensors with the optimal acoustical properties (frequency, beam pattern, etc.) for their particular application: