How google defines links on your site?

[eooc.box]

Here is the situation: I compare links on my site and on site of my competitor in google. I type "link:www.mysite.com" and "link:www.hissite.com" and I see that he has much more links than I. But what is weird, is that almost all sites that link to him also link to me and google doesn't show it in my links. Why?

P.S. I checked the source code of those pages, that link to both of our sites, and I couldn't find the differences between those links. Both links are direct and link to correct URL.
P.P.S. Some of those links are more than 5 years old, so it can't be that google didn't index those pages with links on my site.