The arctic is the region of the world that has at least one 24 hour period per year that the sun does not rise and at least one 24 hour period per year where the sun does not go down. This is known as the arctic circle. As such, the arctic tundra is tundra located within the arctic region. Alpine is that area of the planet that is above the tree line. The tree line is determined by how cold a region gets. I believe that in order for trees to grow, the temperature must not dip below a certain temperature. Hence you will notice the tree line (or Alpine) level differs depending upon the variant temperatures in a particular region. In short, the colder the area, the lower the tree line. In the final analysis, the difference between the two types of tundra is a matter of location. Simplified, basically the alpine tundra is in the mountains and the arctic tundra is like the snowy icy place that we see on TV and think about mostly.