Why do my two analytics tools, incl. goog, report much lower referral rates than a newspaper where I advertise

I am helping a client track their web traffic, especially their online marketing efforts. To track, we are using our own internal tracking system (still in development) plus google analytics and an independent analytics service. The trends being registered on google analytics and the other analytics service are the same, although the numbers are, not surprisingly, slightly different. However, a vendor of the client, a newspaper, is claiming they have sent roughly 4,000 more hits than any of our sources indicate. My goal is to be both impartial and accurate, and naturally the newspaper wants to keep the account and is making the case that our numbers are way off, but the pressure is on to clearly explain why there could be such a discrepancy.

I am wondering why the newspapers numbers might be so high, what the differences in reporting might be, and how to explain all this to the client. I am also looking for good online resources about issues related to tracking online traffic.