There is no difference based on the month the child was born in. There is some other difference in your situation.
Please review the IRS's Child Tax Credit . Do you have several children? Is it possible that you are getting confused about which credit goes to which? For example, when child turns 17, he/she loses the Child Tax Credit but is eligible for the credit for other dependents (which is $500).
Note on the IRS page above that there is a link to the Interactive Tax Assistant to see if you’re eligible to claim the CTC, ACTC or ODC.
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