No tool is perfect and will find all your back links straight away, even Google doesn't find them immediately, so I use a combination of Moz, Ahrefs, Majestic and Webmaster tools to monitor new links.
It will just be a case of when Moz crawlers, crawl the website you have got a link from.
As Moz is relatively small compared to Google, it cant and probably never will crawl the entire web, so it tends to crawl the more larger sites first and crawls the sites which changes frequently more often.
It all depends what you want from the software.
I use:
Moz, ahrefs, majestic, searchmetrics, rankwatch, SEMRush - I use them all frequently and depending what info / research I am doing use the specific software.
Moz is great and has some wonderful features (including this forum - arguably one of the best on the web), but it does have its flaws - its crawl database is relatively small (but its growing 366 million urls was crawled on its last count).
Thanks
Andy