There is NO similarity between these two.
Actually its not google rank its PageRank and its really outdated Google updates it once a year so dont pay any attention to it.
Ahrefs on the other hand is updated more regularly on a daily basis, but still the results can be manipulated and they crawl what they can so some links might take more time to get crawled or they might not be crawled ever. Its a usefull tool that can give you an insight on your backlink profile and can help you solve a few other aspects on your site, such as domain health and other things about your competition.
Dont focus on your "Ranks" as they might drift you from your goals. Focus on your site Content and your page well-structure and then try to promote it.