Which one is better for a BTech course: BITS Pilani or Manipal Institute of Technology Karnataka?

It's highly illogical to compare a Tier-1 engineering college like BITS to a Tier-2 university like Manipal Institute of Technology. 

One cannot compare universities like Manipal, VIT, SRM, Jaypee, Amity, etc. to IIT, NIT, and BITS. These colleges have no brand value of their own and most of the people coming here are those who failed to qualify for competitive exams like JEE/BITSAT/AIEEE/etc. Moreover, the exposure and opportunities offered at BITS are unmatched. It's probably the only private engineering college that enjoys excellent global recognition too. 

Irrespective of the branch you go for, the BITS Pilani degree has similar value everywhere. The BITS alumni network is great with some of them being co-founders of Redbus, Hotmail, etc. 

Here is a brief comparison of these institutes 

The only factor you need to consider while joining BITS is the high fees and academic pressure. So choose wisely!