You should choose MU over Amity for the following reasons:

1. MU has a better crowd than Amity for engineering. People with 3k rank in manipal entrance exam study at MU while people who generally don't get into good colleges are also called for counseling at Amity University. This does not mean that people who are good at studies do go to Amity.
2. Speaking of Infrastructure, MU has got probably the best hostels in India. They fully Air-conditioned with attached washrooms. It's like a hotel room and not an average hostel room. MUJ has one of the best Infrastructure and has won many awards in this field.
3. MUJ is not over crowded. It has decent number of students. You will know most of you batch mates, seniors and juniors.
4. The Manipal group is one of the best in terms of education in India. After BITS, manipal is the best in private colleges.
5. MUJ has grown a lot in just 7 years in terms of placements or opportunities that you will get here are similar, if not more than what you will get in Amity.
6. We have one of the best Fest in Jaipur. We had Sunburn Campus this year and DJ Shaan performed in our college, redbull tour bus( 1st campus in India to host Redbull tour bus), Dunes Music festival ( biggest in Rajasthan) and a lot of cultural events and DJ nights.
7. You will get a lot of opportunities culturally​ in MU. Our dramatics club, music club, dance club and literature club have taken part in many inter-college competitions and have won a lot of them.
8. Almost all of the faculty have published research papers in reputer publications like IEEE, ACM etc. There is a huge reasearch grant of 16 crores for the students to take part in research activities.
9. We have sports team for every major sport you can think of. Our football team is one of the best in Jaipur.
10. MU has hosted a lot of national level and international level conferences. This year we had SAE-NIS, which is a National​ level automobile and mechanical conference. We also had a conference on CS. MU will be hosting SINCONF( cyber security conference) and ICCT ( which is on computation and communication) both of which are international level conferences and will have delegates from US, Europe, Australia​, Egypt, Bangladesh IITs, NITs and other countries. People who are part of IEEE will also be part of these conferences.
11. MUJ has collaboration with many Foreign Universities. Students have been selected for summer research internship at a Singapore Polytech University. We also hosted A swedish delegation, they had come for a project on Image recognition. Every year some students from Foreign Universities send a sem at MUJ.
12. We have few start ups too. Martopolis and Eyrie labs are some of the start ups started by college students. You can join them as interns and learn a great deal about how a startup works and it's culture

In all you will have a lot of opportunities to grow and develop you personality here at MU

Hello

Doing btech from private college is likely same.Main placements occur through mass recruiters but on  the basis of choice given by you Manipal is far better than amity.

Good luck...