Before we look into the differences, please not that I am addressing "Publicity" and "Promotion" with Big P's, if you will. You will obviously want to promote (small p) in the traditional sense, many of the library services and offerings for the library through outreach and advocacy - this discussion is based on the idea of managing large scale Promotions.
Now generate a checklist for different types of items your library will need to publicize. For example, librarians will typically want to make workshops, new collections, new databases, and changes to policies made public. As per usual, every library's circumstances will be different so do not treat this list as if these are the only items to consider.
Next, decide which items will have maximum impact via each avenue. Policies, as an example, are vitally important but not the most conversational topic at the library. Consider posting the new policies to the library's webpage, with a call out signifying new content is available in the "Policies" section. New databases, however, could be publicized at subject specific faculty meetings, through social media posts, and on the ends of bookshelves of related call number ranges. You may also consider doing a series of YouTube videos; I favor the method where all videos on a particular database are organized in a playlist with the first video being an advertisement and introduction, while the subsequent videos demonstrate individual features of the platform. Each video should be as short as possible. As an example, see the videos I created for Statista that we use on our library webpage, and Statista uses as customer support. Each library should have a policy for how they plan to implement YouTube or other video series before beginning to use them.
A "publicity checklist" method allows each individual subject specialist to create a custom set of rules for their own department and be a seminal component of the distribution of information while still maintaining a precedence over methodology and best practices. There are many methods for publicizing library items, but this is a simple and straightforward way to make sure you are maximizing your efficacy with minimal staff time and confusion.
Promotion, on the other hand, is a much more deliberate and concentrated effort on a long term project. Think of the Field Museum as a great example for promotional activities. While there are permanent exhibits at the museum that will always be publicized (think Sue, the Evolution exhibit, the Ancient Americas, etc.) through their website, they spend their big ad dollars on temporary exhibits meant to draw new crowds, and returning crowds looking for something new and different at the museum. Of course, "short term" for a museum is still (typically) a 6-12 month visit. We need to consider what our libraries are offering that actually merit a promotion. Implementing a new ILS or client facing webpage, hosting a traveling art exhibit (or anything that could potentially be revenue generating), building a new library or staging a major renovation, or homing a new special collection are all big-ticket projects deserving of a major promotion plan.
Conversely, new databases, workshops, and smaller events should not be "promoted" in this sense; hence the pivotal difference between publicizing and promoting an event. While we of course want people to attend our workshops, major expenses or opportunities like Edward Gorey at LUMA, should take majority the concentrated and focused efforts for promotion. If libraries focused all of their time and attention on promoting every single event or activity, there would be no time for actual library work. Not to mention, the ideas get crowded in patrons heads; if you tell them too many things at one time they will not remember anything. Academic libraries have it easy - semesters (or trimesters) help break up the year in a very reasonable way. Try implementing a promotion plan for major events segmented by the academic calendar. For example, if you have the luxury of a renovation, use the fall semester to hype up the students about the upcoming changes. Use that momentum to engage students in focus groups on building design. In the spring, post weekly updates about the stages of construction. The following fall, host a series of inaugural events to keep students interested, and thank the donors that undoubtedly made the new space possible. Your promotion plan will be dictated by your event timeline; but strategically crafting an action plan is instrumental in the long-term engagement of your patrons.