I just wanted to know the difference between the Earned Income Credit (EIC) and the Child Tax Credit?

Hi, I just wanted to know the difference between the Earned Income Credit (EIC) and the Child Tax Credit? I just filed and was qualified for the EIC however, I did NOT claim my child as a dependent as the child's father will be claiming her. Just wondering if one parent can get the EIC while the other gets the Child Tax Credit?