Landmark detection and Landmark localization are the terms that have not been defined well. It seems like they are just "created" because someone (scientists) needed them. They are similar to object localization and object detection. After seeing the question I have been searching for finding the contexts that they are used so they can be compared. However, I can say either people confuse them or they are used for the same meaning interchange. Before, proving that I will note one another thing. Normally, localization means to simply localize an object that is, finding the position of an object, or specifically drawing a bounding box around an object. Detection uses localization as a function, for example, in object detection all the objects in the image are localized and labeled accordingly. Thus, it might be expected that if a single set of landmarks (human face) are identified then it would be a landmark localization, and if multiple sets of landmarks (human face, human hands) are identified then it would be landmark detection as in object localization and detection.
So now, check this link where Nvidia names identification of different sets of "objects" as landmark localization. Secondly, here <PERSON> explains exactly the same thing in the introduction of the paper that you provided, but he names it landmark detection. If you search for it, you will see there are a lot of papers that use landmark detection and landmark localization for explaining the same thing.
I don't claim they are the same thing, de jure they might be different concepts but de facto they are used to define the same thing, maybe because they are extremely confused. Thus, it does not "give chance" to understand their meanings.
Update: I asked the question to my Prof. 683
from the university. I will briefly paraphrase the respond: When it comes to localization, it seems to give more importance to location. For example, in a robot application, the robot does object localization or landmark localization to determine the exact coordinates of the object to maybe hold it or touch it. In detection, we actually mean there is something here. For example, I took a photo. There is a tower in this photo. As a tourist I want to know which tower this building is. Finding the tower in the picture can be described more as a detection problem. Another example is a tumor. For example, there is a tumor in a medical image. I might want to localize this tumor in the real body using the image. Or I can detect the tumor in a biopsy image and say that this person has cancer. Although they are usually used for the same thing, you can see one is used more than another in different application areas. 683