IISERs and NISER are reputed institutes in the field of Science. If you want to choose among these institutes, do so based on your choice of course. NISER comes under the Department of Atomic Energy (DAE) and has a wonderful Physics faculty. Among IISERs, IISER Kolkata is the best for Physics. IISER Pune is a good option if you are interested in Chemistry.IISER Mohali is pretty good for Algebraic Mathematics. 

Both institutes are good for research and basic sciences. There is no clear winner between these two. The major difference between the two institutes is that NISER comes under the department of atomic energy and IISER comes under the Ministry of Education. If you are getting IISER Pune, Kolkata, Mohali, go for it. Otherwise, it’s better to choose NISER.