There is no difference between Mercator and a Robinson projection. The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical purposes because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments, despite the fact that those lines may be curved lines of constant bearing in reality.
The Robinson projection is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of representing the globe as a flat surface. The Robinson projection was invented by Arthur H. Robinson in 1963.
There is no right answer to this question. It depends on what you are trying to show. If you want to show the relative sizes of the countries, the Mercator projection is more accurate. If you want to show the relative distances between the countries, the Peters projection is more accurate.