Both IITD Mechanical as well as IIT BHU CSE are good branches.
IIT BHU is 100 Years old Institution and offers a huge Alumni Network. The course curriculum of CSE is designed in a way that will give you enough Industry exposure. Coding Culture at IIT BHU is also very impressive with 10-12 GSOC Selections every year. The Department Club of Programmers (COPS) conducts Codefest which is the annual Department Fest which witnesses the participation of students and programmers worldwide. Moreover, besides Technical things at IIT BHU you can build up other skills since there are many clubs for Extracurriculars like Aeromodelling Club, Robotics Club, Business Club, Cultural Societies, Film & Media Council, etc. Apart from this, the Life at IIT BHU campus and city life of Varanasi is amazing. The average placement package at this IIT is nearly around 8.19 Lakhs while the Highest was 1.54 Crore in 2019. All the top companies including top tech giants Google, Microsoft, Facebook, Amazon offer jobs as well as other sectors where students get jobs are Consulting, Analytics, Education, Business, etc.

IITD is an old well-reputed IIT. Faculties of the Mechanical Department at IITD are very good and experienced. Also, Peer Group at IITD is very competitive since many top rankers join this college. Mechanical Engineering course curriculum is good and one which provides industry skills. Besides Academics IITD too has many Clubs, Societies where students can join to develop different skills. Also, Campus life of IITD and the City life that Delhi offers is quite amazing. Placements too at IITD are good. In 2019 Mechanical Engineering was the brunch with the highest placement records. Many top Mechanical Core companies visit campus such as HUL, ITC, WIPRO, CBS, HONDA, etc. There are also other job profiles available such as Data Analytics, Software, Consulting, Finance, Education, etc.

You should choose the branch depending on your Interest. Choose Mechanical only if you have keen interest otherwise choose CSE to get more opportunities. 

There are different viewpoints for the same question. It primarily depends upon what is best for you, and whether or not the course you are willing to take interests you. 

One of my friends encountered a similar problem likewise, and addressing the same, here are some points you might want to consider before diving into any conclusion,

1. A dual degree in Mathematics, Physics, and Chemistry is claimed as a part of Science, while others are considered in the case of Engineering. You will learn the theories of science instead of the application of technology.
2. In Mathematics, you are supposed to learn the series of higher-order and their respective application. While, in Mechanical Engineering, you will learn more deeply about Engines, Boilers, Automobiles, Refrigerator Systems, and much more.
3. A single degree, like Mechanical Engineering, CSE, EE, Civil Engineering, or Chemical Engineering, can be more advantageous than dual degrees. 
4. If you opt for a dual degree in Mathematics, you have a higher probability of becoming a mathematician or a professor. But, in case you pursue Mechanical Engineering from NIT Surat, then you bear higher chances of securing a placement offer at top-tier MNCs.
5. Your choice should always remain ahead of most others. The one piece of advice is that in NIT Surat, you supposedly cannot change your degree from M.Sc to B.Tech following the 1st year and vice versa. 

You should research the same for BITS Goa and find whether or not the rule is similarly applicable in the institute. This is the basic difference you can find between both institutes. 

IIEST, Shibpur formerly known as BESU is doing really well in all the departments. Whether it is about faculty or infrastructure it is growing steadily and gradually.

But as one wants to take a deeper look in comparing mechanical engineering, EE and ETC, let’s take a look on it:

• The research facility of the mechanical engineering department is quite decent and comparable with EE and ETC.
• It is even better at some aspects like it has research collaboration with some of the renowned foreign universities, some of the excellent facilities etc.
• When IIEST, Shibpur was known as BESU even then the mechanical engineering was considered to be the highest ranking department in one of these three departments.
• Also, one cannot forget to boast of tremendous alumni including a couple of directors of IITs.

• When it comes to placement, the mechanical engineering department has a decent placement record so far.
• ME and EE have almost equal numbers of core placements. And ETC has less core placements but the highest package is grabbed by ETC itself.
• The placement for mechanical engineering is really good though. Core companies such as Maruti Suzuki, CESC, Vedanta, coal India, Tata steel, Bridge and Roof, L&T, elegant marine services come on campus.
• So you can easily expect a very bright future of mechanical engineering graduates from IIEST SHIBPUR.
• After becoming an IIEST the placements have improved and after 5 years it will improve more.
• IIEST, Shibpur has always been known for its excellent core teachings. Also, it has one of the best and oldest labs in the country in EE and Mech which has been improved from time to time.

The college is very good for students interested in any of the three departments as mentioned. All the three mentioned departments are very good in IIEST. The thing that matters is your interest and will.

As far as the placement reports say, placement for mechanical engineering in NIT Raipur varies in between 70%-80%. Although the placements are not as high as IT/CSE branch when compared with mechanical engineering. But yes, there is a most essential perk of job security in the mechanical engineering branch which most of today’s software companies and startups lack.

Now moving from core to non-core campus visitors in the fields of Software & Analytics for all the branches, the package offered is usually very low around 3.5–5 LPA.

• The automobile companies like Maruti Suzuki India Limited, Tata Motors, and Fiatwhich visit NIT Raipur for campus placements all give packages more or less between 5.5–6 LPA.
• Vedanta Resources do provide a better package than automobiles but their package includes a large amount of variable pay which depends on individual as well as company performance.
• So in the long run they are paid more or less the same as automobile placed people get.
• PSUs like Coal India Limited, Bharat Petroleum Corporation Limited (BPCL), Hindustan Petroleum Corporation Limited (HPCL) which visit NIT Raipur are among those few ones who actually increase the average by providing a very good pay scale of 10–12 LPA.

Some companies like BPCL AND HPCL have strict rules of allowing only topper of some branches to appear for the placements. Whereas Coal India does not have vacancies for mechanical every year.

Though, the job openings for Mechanical Engineers are always there. This is a reason why Mechanical is called an evergreen branch. Moreover, placements everywhere on the country’s economic situation.

Don't go for placement packages, it may disappoint you. Just keep a good CPI and be conceptually strong, you will surely end up in good companies like Tata and Maruti.