What is an archive?
An archive as "a place in which public records or historical documents are preserved" (Merriam-Webster). Archives contain archival materials, which "are information objects that serve as evidence of past events" (Peter Van Garderen, 2007). You'll be viewing archival materials during your Rose Library sessions.
Although some institutions might have more distinct meanings for the two terms, "special collections" is often at least related to, if not synonymous with, "archives."
What is a digital archive?
A digital archive is similar in purpose to a physical archive, but the historical documents and objects that provide evidence of the past have been digitized (often by scanning or photography, unless a document was created digitally in the first place) and made available online.
Digital archives are usually created with a goal of preserving historical objects and making them available to researchers.
Being asked to find a "digital archive" is much more specific than just being asked to find a "website."
For more details on these distinctions got to this excellent guide form Sam Houston State library.