Personally, I feel that outrightly calling one college “better” than the other isn’t reasonable. Both IIT-P and BITS-P are highly respectable institutions and getting admission into either is not exactly a walk in the park. IIT Patna might be better than BITS Pilani with respect to some aspects and the same holds true otherwise as well.
So for the sake of the question, let’s try and list out the details.
Alumni/College Culture: BITS-P, being established over 5 decades ago, has a vast alumni base and a well established college culture with various fests and events being held every year.
IITP on the other hand is a newer IIT, established in 2008, and therefore we do lack when it comes to the alumni base.
When it comes to the college culture,I assure you that the college culture at IITP is amazing, with everyone here trying to do their absolute best to make sure that the events and fests are as good as they come.