Which is better-studying at IIST or at new IITs (like IIT Hydrabad, IIT Patna etc.)?

At IIST, most of the students are inclined towards ISRO. Even if you are interested in it, you might lose interest in the latter half of your bachelor's and then may regret your decision because IIST has very limited career options. That being said, look for an institute that caters to a variety of career prospects so that your options aren't limited. 

Considering the new IITs, both IIT Hyderabad and IIT Patna show promising results. I have a friend in IIT Patna and after a long conversation with him, I realized it has a plethora of opportunities both in terms of placements as well as higher education.

The placement scenario at IIT Patna is great and has been improving every year. 

•  IIT Patna has all big shot companies for placements like Microsoft, Amazon, Cisco, Samsung R&D, UHG, Practo, Yodlee, Flipkart, Arista Networks, Texas Instruments, Finisar, Tata Motors, L&T, Mahindra & Mahindra, Bank of America, etc.
• For those who prefer a stable job, many PSUs like Indian Oil, Bharat Petroleum, HPCL, DRDO, Coal India, etc come for recruitment here. 
• If you are among the top 1% of your batch you can get placed in companies like Google and Facebook, the pay scale of which is the highest you can get.
• The small batch size increases the probability of getting more placements in top companies than old IITs with huge batch sizes. 

Prospects for higher education:

Only after you become a part of the IIT family will you understand the true essence of the IIT tag that opens infinite doors of opportunities for you be it higher education or internships.

• My friend secured an internship in Germany only because he had ties with the IIT brand they said.
• Out of 37, 4 had the chance to do foreign internships which is a lot
• The students of his batch even went to top universities like Stanford, CalTech, etc. for their summer internships.
• Every year many students get qualified for the DAAD WISE scholarship in Germany, the Charpak scholarship for internships in France, and the MITACS in Canada, etc.
• As for MS and PhDs, around 18 out of 150 students pursued their higher education in US and Europe. Today they are placed in the top companies like Intel Corporation, Texas Instruments, Intersil Corporation, etc. in their respective countries. 

Having covered all career prospects, I will leave it to you to decide which institute to join.