This is one of the most debatable questions. Both NIT Trichy and BITS Hyderabad are well-known institutions that have their strengths and weaknesses.
Here are a few parameters on which one can compare the two-
If the fee is a matter of concern, I would suggest you go with NIT Trichy. Also, for Mechanical Engineering, NIT Trichy is an ideal place. For a JEE aspirant, NIT Trichy is a dream come true.
My personal preference is in this order-
Both NIT Trichy and BITS Hyderabad are equally good institutions and you can choose either of the two after considering each of the above aspects.