Unable to authenicate after IP changed into URL

Hi there, 

We have Sharepoint 2013, and it has 2nic card, one public and one private IP. 

The private IP is used to talked  to the DC that's in the same private lan. 

We were able to authenticate and get into the sharepoint perfectly fine when it was  IP based, as soon as we changed in into the URL, and access it via URL, it is not taking our authentication at all.  

I feel like it is trying to do authentication from outside to inside, because I don't even see the log on my DC at all. 

Why is this happening by changing from IP based to URL...

Thanks!