Totally different part of the law... for the most part. Wiki and BGG are using it for reference and review so it falls under fair use. On a band's wiki page you can put clips of songs for review purposes, but you can't put the full album up and claim it's fair use. The pages that BGG puts up with links to all the spoilers are close to crossing the line, but they are doing it for reviews, so FFG likely cuts them some slack. They would be well within their rights to ask the reviewers to lower the resolution or watermark the images they use.
Just because FFG puts info on THEIR site does not give anyone the right to use the info and data how they see fit. They would be well within their rights to block hotlinking of images as those images belong to them. If someone set up a table on their front lawn on a hot day with a pitcher of cold water and cups with a sign that said "have a free drink", that does not give you the right to just take the the water and set up your own table on your front lawn and give the water out for free. Perhaps the water was there to draw traffic to their house because they were having a garage sale, and now you are diverting traffic away. With the API, you are not only taking the water, you're basically tapped into FFG's water line and giving out their water for free, claiming "I don't see what the problem is, they were giving it away anyways." They have the right to decide how their assets are distributed. Why do you think GW can refuse to let retailers use their product web pics on the retailers own sites?