The meaning of many terms used in this field - "branding", "corporate identity", "corporate design" - varies wildly between industries and countries.
To me, "Branding" is the overall strategy of building a brand, and goes far beyond graphic design or advertising. Virtually everything a brand does - events, campaigns, the placement of products in markets, pricing - is part of the "branding". The graphic design related part of branding in my opinion is much better (and more modestly) reflected by the terms....
Corporate Design (Wikipedia definition, seems to be especially popular in Germany) or Visual Identity (Wikipedia definition)
meaning a set of visual guidelines reflecting a company's outward appearance in all media. It includes, but is not limited to
- The Logo
- The colour palette used
- Fonts and their use
- Spacing, proportion and layout guidelines
With the main goal of maintaining a consistent and recognizable appearance. I think this is what you mean when asking about the difference between a logo and a branding.
A design manual will define these guidelines, often in the form of an actual, printed manual, with positive and negative examples.
However, the use of this vocabulary will vary, and this purist view is not shared by large parts of the industry :) As you can see in the examples linked below, "brand" and "visual identity" are used synonymously. If you want to get the term right before, say, giving a presentation, ask somebody familiar with your specific field first!
Good examples of design Manuals to look at - these show well that no matter what you call it, corporate design goes far beyond the logo:
German (worth a peek even if you don't speak the language)