As you have not mentioned WHICH campus of BITS you’re getting and what branch you’re getting in both these institutions , I’m going to give you a general comparison.
BITS is a great / very good / good ( depending on the campus ) Tier-1 college . It has a moderate/high placement percentage of 85–95 % ( again campus-dependent ) , and the average salary offered is 7–15 LPA ( core , campus-dependent ). It’s vibrant hostel life , nice coding culture , various extra-curricular activities and excellent non-core opportunities are some of it’s highlights.
IIT Patna is a good Tier-1 college . It has a low placement percentage of 78% , and the average CTC is 13.05 LPA . It is a second-generation ( i.e. , between old and new ) IIT but is developing well and it also has a nice coding culture going around . Also , its fees are only around 30% of that at BITS ( any campus ) , which will eventually save you 16 lakhs over 4 yrs ( even more for an MSc + BE at BITS vs BE + ME at IIT Patna) .
Now , different branches have separate placement drives and those cannot be compared , but in my opinion , go for BITS ONLY if you’re getting CSE at any campus/EEE at Pilani/ECE at Goa or Hyd against a non-core branch at IIT Patna , if you’re getting a core branch at IIT Patna ( CSE/Electrical ) then definitely take that over the same branch at BITS , and if you’re getting a lower branch in both then go for BITS ( because of it’s non-core opportunities ) .