I would always recommend you to go for BITS Pilani EEE over IIT BHU Mechanical. Here are a few points justifying my stance.
Academics: BITS Pilani EEE curriculum is modern and as per industry standards. The branch has a highly qualified pool of professors who make use of modern teaching methodologies and conduct lively sessions. the number of top-tier MS in EE, the infrastructure, and top-notch placements all prove the same. The infrastructure at IIT BHU is deteriorating and the labs are equipped with obsolete instruments bringing down the academic quality of the institute.
Network: BITS has a very elaborate alumni network and gives an edge to IIT BHU.The BITS network (BITSAA) is very well organized and the close-knit nature of the BITSian community is rarely matched by IITs.
Brand: Society in general considers the IIT tag to be of superior quality whereas the recruiters and top universities are aware of the BITS influence and thus it enjoys an equally good brand name. Moreover, the IIT branch seems to have diluted with the increase in batch size and the opening of new IITs. The top recruiters know the difference between old and new IITs and thus the packages offered are different. BITS is known for its cutting-edge infrastructure and no-reservation policy.
Overall, BITS is the best option for you!