NIT Surathkal B.Tech in CSE is definitely a better choice because it's one of the oldest institutes with a rich legacy. The department has lots of establishments in academics as well as industry. There are several well-established student activity bodies and an elaborate alumni base with a coding background.
I am strongly against you joining an integrated program at the bachelor's level because you first need to know the exact domain you are interested in. If you realize that you have taken the wrong branch then spending 5 years will become very difficult. BTech gives you enough time and opportunities to analyze what your interests are you can not decide, after 3-4 semesters, you will be able to judge yourself overall. Also, a B.Tech in CSE from NIT Karnataka with a good GRE score may help you get admission offers of MS abroad.
So go for NIT Surathkal BTech in CSE without any second thoughts.
Both are great institutions in terms of academics and in terms of placements too, but there are some major factors that make IIIT Bangalore better than NIT Surathkal.
But there are few demerits of the college too like -
Life at this college can be difficult which is similar to many colleges in our country, as the environment there is not that fancy like it is shown in our movies but one can get used to it after spending a few months.
The college fest is not that great and interesting as compared to other colleges where the fests are organized in a grand way but this is slowly improving as both the students and the college are putting in efforts to make it better.