Ahrefs and Google both have imperfect backlink tracking mechanisms, though they both offer a good representative view into your entire backlink profile. Ahrefs is notoriously spotty at tracking backlinks (show examples where the backlink is live but not tracked in Ahrefs), and Google released a statement noting that they intentionally peeled back their backlink reporting in an effort to reduce the opportunity to "game" the system, in line with their rollback of PageRank.
This is why we staff an internal team to vet and validate every backlink in your link report before we send it to you.