I for one do believe that Google does track where your traffic is coming from and thus can give better weight to both your site and the site sending you traffic.

And by doing this, maybe that is why when you search google for backlinks to your site, the number is always lower than other search engines and tools.

Backlinks not sending you traffic are just not shown by google?

That may also enter into google's page rank system.
If links are on a page ranked site and not sending you traffic, maybe google does not pass on juice from that site like before?

And links and advertising are different but they work together.
Want links, you must advertise.
To advertise, you must have back links.