I have a lot of friends who are doctors today and have completed their M.B.B.S. from either CMC Vellore or AIIMS. I can easily answer your question.

• Most graduates after completing their M.B.B.S. from AIIMS prefer either settling abroad or aiming for higher education while CMC graduates mostly continue their higher education at CMC itself.
• CMC students are required to do bonds in missionary hospitals and their higher education is their second priority. They provide social services in the form of healthcare facilities in remote areas of the country and contribute largely to society. AIIMS students' priority is usually a high-paying job or an excellent university for higher education.
• AIIMS graduates are more skilled in research and thus tend to publish more than CMC students.
• CMC students have more real-time experience than AIIMS students and possess great clinical and surgical skills. They can transform their theoretical knowledge into practical approaches way earlier than AIIMS students.
• CMC gives more preference to Christians and thus non Christians find it difficult to undergo training from MBBS to super-specialty at CMC.

Here is a video that gives an overview of CMC Vellore

These are some major points of comparison I have observed between my graduate friends from the 2 institutes. 

CMC Vellore is better than AIIMS. When it comes to Clinical and Surgical skills, CMC graduates are well ahead of their peers. As a student of the institute, you will get a lot of hands-on experience. Students are allowed to shadow postgraduates for clinical work from their 2nd year of MBBS. This helps gain insides into real-world practice, early. 

CMC has a good international reputation and a strong alumni network all across the world. 

CMC has a few cons as well. Most CMC graduates need to do bond in mission hospitals. So, their higher education gets delayed. Also, it’s difficult to go through the MBBS to super-specialty at CMC, for a non-Christian without mission hospital backing. 

Both CMC Vellore and AIIMS are reputed institutions. One of my friends studied at CMC Vellore, so I might be able to help you with the question. CMC has a fair share of advantages and disadvantages. 

So, here’s stating some facts about it, you can analyze and then take your next step.

• Out of 100 MBBS students at CMC Vellore, 80 are minority candidates referred by various churches across India.
• A 2 years service bond right after MBBS course is provided but your certificate will be held until the bond is completed.
• Learning the Tamil language is mandatory. 
• An Unreserved (General) category candidate will remain a minority in the institute, but your religious views are respected.
• No other medical institute matches the variety and the patient load CMC encounters. 
• Their clinical experience tops any other college across India.
• CMC is internationally renowned and is second only to AIIMS Delhi, whose degree allows you to practice in a few Asian countries like Singapore without any license exam.

Thus if you choose to study at CMC Vellore, it is a great option. But keep in mind that they have a majority of reservations for Christian students, so the competition will be equally tough.