Why do BITS Goa students equate their college to BITS Pilani or IIT Bombay?

If you look at a general perspective, both IIT and BITS are extremely prestigious Engineering institutes in the country. Both have managed to contribute largely to the world's technological map. IIT JEE is the national-level exam for Class 12 students for admission to IITs while the entrance exam to BITS is the only engineering entrance exam in India to evaluate candidates on the basis of Logical Reasoning and English language skills as well.

I have drawn a general comparison between the 2 institutes on the basis of some common aspects

Entrepreneurs: The success of an Engineering institute depends primarily on its Entrepreneurship culture. Some Famous entrepreneurs from both institutes are as follows 

IIT Entrepreneurs:

• Dipchand Deep Nishar was the highest-paid manager at LinkedIn till 2014. Currently, he is a board member of TripAdvisor and a software company called Opower and also works as an advisor for several tech companies.
• Nikesh Arora, a graduate of IIT-BHU, is one of the highest-paid employees at Google. He is the Chief Business Officer and a Senior Vice President in the company.
• Soumitra Dutta from IIT-Delhi is the Dean of Cornell University’s Management School.
• Vinod Khosla from IIT-Delhi again is the co-founder of Sun Microsystems. 

BITS Entrepreneurs

• Baba Kalyani is the Chairman of Bharat Forge.
• Prithviraj Chavan is the Chief Minister of Maharashtra.
• Rakesh Kapoor is the CEO of Reckitt Benckiser.
• Sanjay Mehrotra is the President and Co-founder of SanDisk.
• Sunil Duggal is the C

Placements 

We can safely conclude that placement scenario at IIT Bombay is better than BITS.

Campus life: BITS Pilani has a better campus life and culture than IIT Bombay. It boasts of a well-planned campus, libraries and hostel facilities. It has a unique concept wherein students take tutorials from their various teachers. Its cultural event Grubs is quite popular in engineering students.

Fees: IIT Bombay Total fee is around INR 9,50,000 (BTech) whereas BITS semester-wise fees is 2,31,500/-

Since IITs are run by the government, studying at IIT is cost-effective whereas BITS are expensive.

Infrastructure: Both IIT campuses and BITS Pilani campuses have top notch infrastructure. However, most of the new IITs lack a permanent campus and are also facing a lack of a good faculty positions. BITS has a limited number of campuses that are well-established and it is known to upgrade its world-class infrastructure on a regular basis keeping up with changes in technology.

Faculty: BITS Pilani attracts the best faculty from India and abroad. It offers better salary and more freedom to its teaching staff. Whereas IITs often face faculty issues. Many new IITs depend on guest faculty or visiting faculty to conduct their classes.

I would suggest you to choose IIT mainly because of low fees and the IIT tag.