I've been reading through R's Affy manual, and it refers to other vignettes. Does the difference between these two terms simply relate to quantity of content, or is there more to it?
The reference manual of a package is a single document beginning with the package description and containing all of the content from the .Rd help files for the packages. Generally, this means it has the help files for (exported) functions in the package, any documented data sets, and package-level documentation (if included). It is automatically generated from the .Rd sources.
Every package has a manual. Even a package with no exported functions and documentation would still have a manual (when built) consisting of the text from the Description file.
Vignettes are free-form documents. Generally, package authors use them to demonstrate the use of their package. They are optional, some packages have several (as I write this dplyr has 8 vignettes) and many packages have none.
As mentioned in the comments the R Extensions Manual is the definitive source for all things package-related. Here is Josh's link to the Vignettes Section, and 2.15 Processing documentation files describes how reference manuals are built.