Why do not listed backlinks inside Google Webmaster tools and backlinks in Ahrefs backlink checker do not match at all?

Hi Friend,
They will eventually show.

How recently did you get these backlinks? Some backlinks index almost instantly and some take a very long time to find. There are some ways to speed the process up such as adding all the links you want to index to a Blogger post. But if you want them to index entirely naturally then you can just be patient.

This data comes from Ahrefs' industry-leading backlink database which contains 212 BILLION pages. ... You can use the Backlinks report Site Explorer to check backlinks to any website or web page. So if you enter a competitor's site, you'll see all their backlinks.