As the others have said, you'll never see all the backlinks with one tool... or even more than one tool for that matter. I use both Ahrefs and Majestic. They very rarely match. The issues you are having is why some use another tier of backlinking. The more backlinks to a site, the more liking it will get the once over. Personally, I haven't done a lot of 2nd tier backlinks as the number of 1st tier backlinks has worked for me so far. Granted, my keywords haven't been very competitive.