Hi,
I'd like to answer this as simple as possible. I understand both tools have their own audit section that decides DA for Moz and DR for Ahrefs. The only common thing is backlinks from referring domains. The more high-quality link you get for your website the more chances are your DA and DR will increase. I used both tools and what I see is MOZ fetches the backlinks very late as compare to Ahrefs.
A real example: My domain (marketinglad.in) has 15 referring domains as of now. From Ahrefs, I can see the same number in referring domains and from Moz, I see just 4 referring domains and that's not a piece of correct information. That's the reason My DR is 17 and My DA is just 5.
I hope this helps.
