There is no role of place in vaccination
. You can vaccinate your child anywhere. Just make sure that you have got the immunisation card, which gives the information of the vaccines taken earlier.
In govt. PHC vaccination is done mainly according to national immunisation schedule.
Now in pvt. setup physicians usually vaccinate according to IAP schedule which covers more vaccine preventable disease. So as you are planning to vaccinate in pvt. clinic you will get an opportunity to get your baby vaccinated for some more diseases like rotavirus
, pneumococcal and hemophilus.
Regarding efficacy there is no difference.