Hey there! Tawny from Moz's Help Team here. I think I can help explain why you're not seeing all the links you might expect, and why you're seeing some older links that you're not seeing in your main Campaign Site Crawl.
Open Site Explorer and the Link Analysis page of Moz Pro Campaigns are both tied to our Mozscape index, which tends to update roughly once a month.
Just a few points on how we compile our index:
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We grab the most recent index.
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We take the top 10 billion URLs with the highest MozRank (with a fixed limit on some of the larger domains).
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We start crawling from the top down until we've crawled ~130 billion URLs
The idea here is that we're focusing on the highest-quality links we can find, coming from the most prominent pages of authoritative sites. So, while you may not see every link for a site within our index, we're aiming to report the most valuable ones available!
Most new sites and links will be indexed by our spiders and available in Mozscape and Open Site Explorer within 60 days, but some take even longer for many reasons, including the crawl-ability of sites, the number of inbound links to them, and the depth of pages in subdirectories. This tends to bias our index in favor of newer links. Linking data is only stored in the index for about 180 days, so unless the crawler has a compelling reason to return to your site and the sites linking to yours and rediscover those links, they can fall back out of our index again. That doesn't mean they're not out there affecting your SEO, just that our tools don't see them anymore.
You can see our most recently updated schedule here as well as some more technical metrics on our Mozscape API Updates page. You can also see when the last and next updates happened on the Open Site Explorer (OSE) homepage at any time.
Since Moz focuses on quality of links over quantity, we are always focused on the most relevant links to display to our users. It's possible that Moz's index will leave out some of the lower-quality (non-link juice providing) links out of our index because of this. So, that might explain why you may see some discrepancies with what other tools may be showing.
You can read more about how we build our index in our guide here.
I know this is a ton of information, so if you have any questions or if I didn't make anything clear enough, please don't hesitate to ask! You can always drop us a line at help@moz.com and we'll do our best to clear up any questions you might have.
Cheers!