Which one should I choose, BITS Pilani for mechanical or IIIT Allahabad for IT?

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.