Firstly, you should know that there is no difference in the facilities and reputation of the colleges, Although BITS ranks better as an engineering school. BITS Pilani ranks #30 on NIRF's Engineering rank list and IIIT Allahabad ranks #103 on the same. But you should focus on your monetary capacity in consonance with your future plans before applying. 

My cousin studied at IIIT A and the following points are based on his experience:

• IIIT A is an excellent place to study coding. The classes are good although the syllabus has to be maneuvered because it's tough. You can manage it through attention in the classroom.
• Attendance is an important part of the academic process here because IIIT A has a 75% mandatory attendance rule.
• Without a doubt, BITS has a higher fee range as compared to IIIT A, about 7 LPA for a 4-year program.

Here's a table comparing both institutes:

Lastly, if you love the mechanical stream and are truly motivated to study Maths and Physics then you must go for it. Mechanical is a vast field as well because you'd learn everything related to mechanical processes and even production scheduling. So clearly check your interests and goals before deciding.

Yes, it was confusing once, two years ago and now I can happily confess that I do not regret choosing IIT Gandhinagar over BITS-Pilani. Since I also have friends in both the colleges, I can provide a detailed overview of the good and bad aspects of the colleges.

Also, information on BITS might be limited due to it will depend mostly on other people’s experiences, but the answer will talk about the good and bad side of Mechanical Engineering at IIT Gandhinagar. 

Following are the merits of IIT Gandhinagar:

1. Limited batch- The class consists of 30 students. This makes the interaction simple and competition healthy. The classes are interesting and interactive. And BITS here, has a classroom filled with 100 students. 
2. Friendly professors- All the professors are cooperative. They will always be present to help you. 
3. Constant branch parties/meetings- The Mechanical Branch will always witness regular meetings wherein the latest course will be the topic of discussion and how things can be done to modify it and gather suggestions accordingly. These meetings include Mechanical Engineering students (no freshers) and professors. 
4. New course- In case 10 others from your branch decide to go for an entirely new course which isn't taught in the department- you can always consult the professors and if they can teach that, your will certainly be granted. 
5. Lab facilities- Your lab experiences would be such that you'd work under time-limits. Objectives that will be given to you would have to be completed under a time-limit present along with it. 
6. Inter-disciplinary ideas- Unlike the rest of the colleges, IIT Gandhinagar has neighbours who are from a background of MSE and CSE. While sharing ideas, you get to learn much more than you can imagine.

One thing that students could miss at IIT Gandhinagar is the cultural fests and events. They don't have grand cultural fests of their own. This privilege was taken away by the seniors, which we were trying to rectify. But gratefully, we already have enough, so, that doesn't make us expect too much of these grand fests for ourselves.