My brother is currently studying IT at IIIT Allahabad and according to his first-hand experience and your interest in IT, the order of preference for you should be:

IIITA>MNNIT>BITM>MANIT

It is because if you are genuinely interested in IT then IIIT Allahabad is the best option you can choose because of the following key points:

• The campus is well equipped in terms of infrastructure and facilities and the coding culture is quite good there. Also, it is an institute specifically made for IT, so the infrastructure is planned according to it.
• The course structure at IIIT Allahabad is also strikingly different from what you'll find at any other engineering college in India. It focuses a lot on programming and electronics-related subjects, right from the first year itself.

In NITs, you need to study a lot of subjects not at all related to your branch or course!

However, MNNIT or BIT Mesra may prove to be better options in case you realize that computers are not your cup of tea and decide to go into the non-tech sector. You will have a lot of diverse companies visiting these campuses, while in IIIT Allahabad you’ll see only IT companies coming to recruit.

Also, JEE Main rank of all the options you mentioned will differ from each other. So your rank also plays a major role in selection.

There are few differences between the CSE and the IT branch in MNNIT in Allahabad. One of my friends studies at MNNIT, so I can give you the basic difference between these two branches.

1. Academically: 5th and 8th Semester subjects of CSE and IT branch are different. 
2. Learning: There is not much difference on the learning basis. Core subjects are the same in both departments. In both departments, you will learn through self-study. 
3. Placements: There is no difference in placement of CSE and IT branches. Several companies come for placement for both departments, giving you several opportunities. 

The recruiting companies are more or less the same that offer both CSE and IT. The list includes - Microsoft, Google, Accenture, Oracle, PWC, Capgemini, Byju’s, Jaro Education, Adobe, Goldman Sachs, etc.

There is only a slight difference between both departments. Both the departments of MNNIT Allahabad are almost the same on the basis of learning, placement. So, if you want to join MNNIT Allahabad CSE or IT branch then go for it.

CSE at IIIT Allahabad wins hands down when the choice is between IIIT-A and BITS Goa. Both institutes have the same coursework for CSE, but IIIT-A offers it for one-third the fee of BITS Goa. Most of the students opt for BITS just for the brand name. The faculty at the BITS campus is not as good as that of IIIT Allahabad as it is funded by MHRD while BITS is a private institute. 

IIIT-A is popular for its coding culture. IT is one of the flagship courses offered there and placement records are at their best. Placement in the software domain is any day better at IIITs when compared to BITS. 

The following table shows a brief overview of the institutes:

IIIT Allahabad is the best choice among the two with good placement records, perfect campus, amazing faculties, and facilities. Being a government-affiliated institute has its own merits when compared to BITS Pilani Goa.

IT-Business Informatics at IIITA is dominantly a B.Tech IT program with all compulsory subjects of B.Tech. IT is already a part of the B.Tech. and so is the IT-Business Informatics program. The Program while maintaining its core IT structure keeps whatever is mandatory for a Technology Bachelor program as per national statutory bodies specializes at Elective subjects level with Business Informatics specialization as a rather essential part of the program.

• The first Semester is exactly the same for all the branches and all the specializations with an option of internal sliding as per Institute rules.
• The subjects with Business Informatics are included in B.Tech. IT program as specializations in the consecutive semesters with other associated activities like Summer Internships, Project work, etc.

Also, the new program is incomplete consent with national statutory bodies and thus all the opportunities available for B.Tech. IT program shall continue to be available to the B.Tech.

IT-Business Informatics program as well including that of higher education and job-related requirements/skills etc.

Yes, you will need a laptop for the 1st year of IT at IIIT Allahabad. Here are the reasons why. 

• As there is coding learning in your course, you will require it.
• If you can’t manage to buy all the books(or get them from the library) or are not comfortable with reading from e-books on your smartphones. You should get a laptop.
• Can you run to the computer labs about a kilometer from your hostel any time you face a question or something tricky? Or would your friends be willing enough to allow you to borrow their laptop whenever you want? If the answer to these questions is no, then certainly it would be important for you.
• And if you have interests in fields other than coding then definitely you should bring it.

I think these points are enough for you to decide.

A semester-wise view of the syllabus is discussed below. 

Semester-1: 

This is the general semester, where you have to study subjects like Physics, Chemistry, Engineering Drawing, Computer Programming language, etc along with labs and tutorials.

Semester-2:

In this semester you study advanced versions of subjects like Physics, Maths, Chemistry, Ecology, etc along with labs and tutorials.

Semester-3:

• By the 3rd semester, you get into the department and then you get to study your branch-related subjects.
• In this semester the courses are Data Structures, Discrete Mathematics, Electronics Signal Processing, and other IT-related subjects.

Semester-4: 

• As the 3rd and 4th semesters are crucial, you are taught major stream subjects in-depth such as Algorithms, Graphs and Combinatorics, Automata Theory and Computer Organization, etc.
• There will be labs and tutorials related to the courses in the 4th semester.

Semester-5:

Here you learn important courses like Operating Systems, Computer Networking, Cryptography, and Object-oriented modeling, and computer graphics.

Semester-6:

• In this semester courses such as, Database management, scientific computing, multimedia technology, Software engineering, etc are provided to the students.
• Also by the end of the 6th sem, you are allotted the mini-project.

Semester-7: 

• In this Sem, you get to opt for electives.1 open elective and 1 professional elective choice. And other subjects like Digital Image processing, Professional Ethics, Service-oriented software Engg are taught.
• These subjects are very easy and quite scoring. But you will need to work a lot on the project. 

Semester-8:

• In this semester, the department offers 1 open elective and 2 professional electives
• Apart from these, the courses are Privacy-preserving Data publishing, Research Trends in IT.

You should start coding and learning the basics of programming languages on your own. This will help you get prepared for internships.

During your years in MNNIT, I would advise you to maintain good pointers and pay attention to coding and competitive programming. Score well in subjects and try to focus more on highlighted courses because companies ask questions from those courses. Also, make good projects.