There are several tools that you can use for the backlink analysis. As you asked the best tool between Ahref and the Moz so both are great tools for the backlink analysis. If you are looking for the paid versions of these tools then Ahref is more expensive then MOZ and gives you the indepth report about the backlinks pointing to your own site and you can also see the backlinks report of your competitors. Moz shows you where your backlinks are come from, spam score, follow or no-follow.
Ahref is recommended as it is widely used for the backlink analysis but it is expensive.