Two of my cousins went to BIT Sindri and BIT Mesra, Patna extension.According to both of their experiences, BIT Sindri is better than BIT Patna.Here is a comparison of the two institutes. 

Fees: 

One of my cousins spends one lakh per semester for his BE degree at BIT Patna. While the other completed his BTech degree for around 70,000-80,000 at BIT Sindri. 

Type of Degree:

BIT Mesra offers BE degrees for engineering. While BIT Sindri offers BTech degrees for the same. There’s a slight difference between BE and BTech. BTech degrees are only delivered by Govt. colleges like IITs, NITs, etc. and BTech has an edge over BE.

Infrastructure:

BIT Sindri definitely offers better infrastructure despite the fact that it is quite an old institute. 

Hostel Accommodation:

The hostel accommodation at BIT Sindri is not good. BIT Patna has much better hostel facilities.

Other Facilities: 

Playgrounds, hangout spots, Wi-Fi facilities are far better at BIT Patna. 

Placements:

Placement at both institutes is almost the same. My cousin from BIT Sindri was placed at Electrosteel. Whereas, the one from BIT Patna was placed at Infosys. 

Campus Life:

Well, campus life at all the engineering colleges has the same pattern, i.e. friends, parties, fest, etc. These institutes are not the exception. 

This information should be enough to choose one of these two institutes. So, choose according to your preference.