What are the benefits of an oral exam?
Pedagogical advantages:


It is much easier to catch misunderstandings early and thus "rescue" an answer. After all, bad exam questions aren't that rare (bad as in: if the student has a very good understanding of the subject, they may be able to guess what topic the examiner has in mind).
Misconceptions can be corrected: while a written exam gives a snapshot of what the student understood or not, in an oral exam the examiner can ask the student to think again if the answer is wrong. Or can give a counterexample that takes into account the student's answer
In the end, an oral exam can be a discourse on a subject, which IMHO allows for better/easier grading. 
Related: the difficulty can be adjusted during the exam according to how much the student knows. 
Many oral exams I had had a "mixed" approach for deciding how deep into each subject to go: often 2 subjects were covered in depth (one by choice of the student, one by choice of the examiner) and a number of other topics touched. Particularly letting the student choose a subject of their liking (usually as the beginning of the exam) is not possible with written exams. 


Practical consideration:


if only few students are to be examined, oral exams take much less time for the examiner. 


Disadvantage:


If the examiner is somewhat off in estimating the difficulty of their questions, a written exam at least has the same questions for every student.