Should I take NIT Surat for mechanical engineering or BITS Goa for an MSc in maths?

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.