This is very difficult to differentiate, as both of them are hugely notable for Neurosurgery. But like every other institution, they also have their fair share of pros and cons. Let me discuss some facts about both the institutes, analyzing which you can make your final decision.

• AIIMS has given many good Neurosurgeons. It has a high credibility value.
• It has got a dedicated trauma centre, cadaveric training and research facility, neurosurgical education and training school, and animal dissection lab.
• NIMHANS is a dedicated mental health and neurosciences institute. 
• It has got NBRC - Neuro-Biology Research Centre which provides proper infrastructure for research and technology advancement.
• There is no difference in the quality of work, patient management protocol, etc. at both institutes.
• The only thing bothering the students of AIIMS is the increase in faculty members. This divides the surgical work and less share comes on part of residents.
• NIMHANS has fewer faculty members as compared to AIIMS.

You can also choose the college favoring your location. North Indians generally opt for AIIMS while South Indians will go for NIMHANS.

Ultimately, it depends on the student’s choice, because both the institutions are equally good. But a university name does not matter, it might give you the first push but after that your training and your work will speak for themselves.