Moz is so slow to update I have a site that has been stuck at the same rating for them for 6 months while majestic and Ahref have updated it ...
Moz is only good for SERP tracking, keyword search and local SEO is not bad they have some nice tools there really their forums are the only thing worth a crap once you have set your site up and just to get one more perspective ... their site is really only their to give them more authority so they can get big contracts like the one with Yelp.
Ahref is the best at updating backlinks as Google makes it so challenging to do since you have to download a spreadsheet and then run a micro to tell you what is new and what is gone - if you are handing with excel you can replace Ahrefs - their world rank based on backlink chart is free to access so no need to sign up and it take it with a grain of salt as it has some strange swings from time to time and does not really reflect your SERP BUT it is a nice indicator.
Majestsic - I have to be honest I do not use it as much the few times I have checked it with the free version Ahref had more backlinks and that is the only reason I can see to use it with my limited time - maybe someone else can chime in.
So summary for me use Moz for a free trail first 90 days to get your keywords charted, rank your serps, look at local, etc
Then sign up for Ahref when the backlinks start rolling in and monitor them and your ranking for those keywords.