It's not the very same logo. Both use a Polar azimuthal equidistant (PAE) projection - the North Pole is the center point. The FES uses a 19th century map with South America at the 6 o'clock position. The UN originally (1945) used a similar orientation, with South America in the 6 o'clock position, but a couple of years later (1949) gave the map a quarter turn so that Africa was at the 6 o'clock position; this change undoubtedly had political significance.
An Azimuthal Equidistant projection is very popular as a decorative world map, even though shapes and distances are much distorted nearing the edges. It is used, with different cities as the central point, for example, in broadcasting and shortwave to show which direction antennae should be aimed to send/receive with various target countries; also with aviation to indicate distances direct flights would require. This projection is even more often used with maps of much smaller areas, such as states, to indicate driving distances from the center point.