One of my cousins is a student at IIIT Vadodara. I hope I will be able to help you with this doubt.
BVM is an old college whereas IIIT Vadodara is very new to its comparison. What my cousin says is, if he had a choice between BVM and IIIT Vadodara, he would have definitely chosen BVM over IIIT Vadodara.
IIIT Vadodara provides you with very few resources. There is no permanent campus and it lacks hostel facilities. It would not be surprising to see that it still lacks these two basic infrastructures even after ten years from now.
My cousin, who is in his fourth year of graduation, regrets joining IIIT Vadodara. Although the tag of an IIITian may help you in getting a better placement than BVM.
If you are getting an opportunity to opt for BVM, go for it. It is a well-established college and you will get ample resources to utilize for your growth.