What should I prefer, IIIT Delhi for ECE or BITS Hyderabad for mechanical?

I will suggest you consider your own interests first rather than the placement scenario. For this, you can go through the course structure of both branches. If you still are unable to determine the right choice for yourself, go through the following pros and cons of both institutes

IIIT Delhi 

Pros:

1. IIIT Delhi has a more Research-oriented curriculum and has an interdisciplinary approach to the curriculum.
2. The packages offered are at par with most tier-1 colleges like DTU, NSUT, decent NITs, etc.
3. The lab facilities are modern and top-notch

Cons

1. IIIT Delhi has a very small campus.
2. The college is relatively new so the alumni network is very weak.

BITS Pilani

Pros:

1. Excellent noncore placements especially management/finance.
2. The curriculum is very flexible and students of all branches get to choose minors in Data Science, Finance, Robotics, etc. 
3. The college has a huge brand name and alumni pool which adds credibility in promotions and pay grades.
4. The college has the PS system, which gives every student 8 months (6+2) of internship in top companies solely based on your preference and CGPA. 
5. The institute has a very charming campus life

Cons:

1. No compulsory attendance policy which might affect those who aren’t very self-motivated.
2. A lot of participation in festivals and cultural events is required which although good, might affect your academics negatively. 

Ultimately, choose what signs with your preferences.