There are several reasons for this discrepancy:
1.) No backlink crawler can depict the whole internet.
The number of found backlinks depends on the time since when the crawler is active, as well as on the fact wether it has a certain focus , f.e. a geographical one.
2.) Sitewide links: saved only once in linkbird, because it does not make sense to monitor them all.
3.) linkbird only saves links that are online
Especially 2.) and 3.) are important when comparing linkbird to Webmaster-Tools. Google Webmaster-Tools puts out all sitewide links and also links that are offline.