Hi OP!
Great question to ask and there is a lot of depth behind your question.
So to start off, none of these 3 sites "build links" but rather help you find, discover and gauge their authority.
From my experience Ahrefs has by far the biggest index and finds the most backlinks than most 3rd party tools. They also offer competitor backlink research to help you find possible link opportunities.
Majestic is also a great tool for evaluating the metrics of a domain, many use it when looking to build out a PBN or when buying an expired domain. Their metrics are called CF and TF, both of them very popular terms and methods used around the forum to evaluate the quality of links and domains.
In regards to Moz, they are mainly good for evaluating a domains authority metrics like Majestic except they have their own metrics called DA / PA.
In a nutshell even though all three sites seem to have similar offerings, each tool has specific features that the others dont. For best results I also suggest using a combo of all three if you are running an SEO agency or are trying to rank multiple sites as they subscriptions can be quite high!
Best Regards,
Mark T. - President