I downloaded your tileset and everything looks fine for me both in the editor and during runtime. Which version of Godot are you using? I'm using version 2.1 on Linux.
Taking a close look at your image, you have two versions of the image. Are you using the first version, or the second. If you are using the second version, is the filter checkbox disabled there?
Also, does disabling mipmaps change anything?
Here's a link with a explanation of why textures/images should generally have a image size of the power of two.