It is totally upto you. If you want IITian tag go for IIT patna because its bitter truth very few person know about NIT. In abroad no one know about even NIT trichy.
Next if you want to get placed in better company with better packages then opt NIT patna CSE which have avg package is around 9–10 lpa. AVG package of ME at IIT patna is around 6–7 lpa and very few company visit IIT patna for core branch. Many people in IIT in starting opt core branch for IITian tag but in last year they study CSE for better placement. Then why not i should opt cse at NIT patna and study CSE from starting.
In brief if you have interest in ME then opt IIT patna.
if you dont know your interest then opt NIT patna
in location wise IIT patna is not good it is situated in bihata