Comparison of methods
Elastration is a procedure where a very tight rubber band is placed around the scrotum (or tail), cutting off blood flow and causing tissue death. The band is usually left in place until the scrotum (or tail) drops off. This is a non-surgical method of castration.
In surgical castration the scrotum is cut open and the testes are removed from the body. The wound is usually sprayed with a topical disinfectant and heals without requiring stitches.
Risks of both methods
Elastration causes pain for several days, which in turn can activate innate immune cells, directly causing an inflammatory reaction and increasing the risk of infection (source). If the animal subjected to elastration doesn't receive any anaesthetics or if the anaesthetics wear off before the procedure is completed, they have a higher risk of developing an inflammation (source).
Experiencing severe pain during during the first days of life can also influence how the animal will react to pain later in its life. In an experiment, female sheep that had their tails docked by elastration neonataly seemed to experience more pain and stress while giving birth later in their lives (source).
Surgical tail docking is known to cause significantly greater acute pain responses than the former [elastration or docking with a hot iron]. (source)
Surgical castration is the most common method of castration in calves and the most common complications are swelling, stiffness and increased lying time (source).
Comparison of complications for both methods
This study found no difference in mortality rate between uncastrated and castrated lambs, regardless of the method of castration.
In this survey vets reported more complications after non-surgical castration (by elastration, clamping or other methods) of calves than after surgical castration. Most non-surgical castrations of perinatal calves were executed by the producer, which might influence the outcome. The most common complications after non-surgical castration were swelling, stiffness / altered gait, recumbency, wound infection and anorexia. The most common complications after surgical castration were swelling, stiffness / altered gait and hemorrhage.
In this study surgically castrated lambs showed significantly more stress signs than non-surgically castrated lambs, even after receiving the same type of anaesthesia.
Elastration seems to cause more physical complications (like wound infection), but surgical castration seems to cause more pain.
The younger the animal is at the time of castration, the less complications were reported.
In all the studies a non-surgical castration by the Burdizzo method (crushing the spermatic cords) seems to cause the least pain and the least complications.