This is a mipmapping issue.
When painting the texture, Blender disables mipmapping (so everything looks shaper) so you don't notice the problem.
When you view the object from a distance, it's actually using a scaled down version of the texture, which shows the black background bleeding through, hence the wires. The solution is to increase Bleed.
You can also fix it in a 2D software using the 'maximum' filter.