As pointed out, they are certainly not the same, but it might not be a bad idea to track and report on the direct traffic. If there has been outreach done and the company is mentioned in print with a URL, direct traffic (along with some search traffic on the URL or business name itself) is likely to go up. If your email newsletters are not tagged, they're likely to show up under direct traffic. Depending on your role, some of what you do under the greater SEO/inbound marketing role can show up under the direct traffic.
To give you an idea of just how much money is being spent on paid search, take a look at Google. Google's AdWords program is the most used pay-per-click (PPC) advertising program available today. While the tech giant owns YouTube and Android, among hundreds of other profitable brands, AdWords accounts for roughly 70% of their revenue -- which speaks wonders for its effectiveness.
At our agency, we work with sites of varying sizes, from very large to quite small, and recently, we have noticed a trend at the enterprise level. These sites aren’t relying as much on Google for traffic any more. Not only are they not relying on Google traffic, but also, the sites are getting less than 10 percent (or slightly more) of their organic traffic from the search giant.