Answer:
A visit to a specialist most likely becomes handy when the medical condition is critical. Critical in the sense that the generalist medical practitioner has tried unsuccessfully to treat such a condition without success.
For instance, I know a friend who has tried severally to treat as skin allergy which occurs on her hands and fingers. At first, we all thought it was a minor skin irritation or fungal infection which would go away. She became worried after seeing three different generalists doctors and using at least 8 different epiderm creams.
An epiderm cream is a cream that is used to treat skin infections or conditions.
So at this point, she decided to see a specialist.
One of the advantages of seeing a specialist is that they are very versed in the treatment of a narrow area of medicine and are most likely to come up with solutions that were missed by the generalist. I once had an eye allergy which was recurrent. I was forced to see a specialist after several years of managing it and being used as a "guinea pig" by generalist practitioners, all of which were unsuccessful at treating the ailment and who didn't have the decency to recommend a specialist.
Another advantage of seeing a specialist is that they are thorough and unlike the generalist, less likely to give a misdiagnosis. A misdiagnosis or a wrong treatment, such as it was in my experience above, can prove to be quite costly and even fatal.
So it's best to see a specialist if you have tried a generalist medical practitioner and the situation persists after several treatments. A good generalist practitioner would recommend seeing a specialist as soon as they spot a history of recurrent unsuccessful treatments from generalists.
On the other hand, for treatment of minor ailments, such as a cold, a minor burn, or infection, it is safe to see a generalist.
Cheers