They aren’t the same thing.
Branding is the process of creating a memorable relationship with an audience by reaching out to them and increasing the brand’s visibility. Branding efforts usually take the form of a story and contain engaging material, but not any type of direct sales pitch.
Direct marketing, on the other hand, is the process of interacting directly with a target audience in order to persuade them to make a purchase choice.
Running an interesting blog on a corporate website and distributing it across social media accounts, for example, is an attempt at branding. The goal of the blog is to bring visitors back to the website on a regular basis, provide them with material, and establish a relationship with them. Running PPC ads with Call To Actions like “BUY NOW” is, on the other side, an effort at direct marketing.