Credit Bureaus give credit score which signifies the credit worthiness of applicant. This score is result of various factors such as repayment of loan on time, period of credit history, quantum of credit facility utilisation etc.
CIBIL, Experian or CRIF scores you are talking about are credit scores issued by different Credit Bureaus.
In India, layman calls CIBIL as credit score. But, CIBIL is just another credit score by a Credit Bureau.
Similarly all these are credit score provider agencies.
Banks normally verify CIBIL score, as it most prominent credit bureau in India.
Often banks refer score from different credit bureaus, as sometime due to some errors your credit score might be wrong or misinterpreted.
There is no difference between CIBIL, Experian or any other credit score, but only difference is that they are provided by different Credit Bureaus and few factors to decide score vary.