My directions are in a mishmash of GIMP/Photoshop-ese at the moment, since this isn't a formally written tutorial. Pardon any sloppiness. I can clarify a section if required, and make sure its in proper photoshop-ese.
First of all, with this map, it would be easiest to make it wrap around north-south. If you wanted to make the east side wrap around to the west, then you'd have to put in some significant time drawing.
Offsets:
http://www.cadtutor.net/dd/photo/seamless/seamless.html
For the graphical component, I suggest setting an offset. This way you can see how the edges will match up. For the first round of work, just use the color picker to select the border color, and then select a slightly fuzzy brush and redraw the province borders to your preference.
Keep in mind that provinces are determined with a single pure white pixel, if you decide to join two provinces.
Since this is only a random map, we're going to take a few shortcuts and be a little sloppy. When the seam is obvious, but the terrain is similar, try using a smudge tool, and dragging it accross the seam. Alternatively, you could try a blur tool. When the terrain along the seam differs wildly, try using the clone tool.
This won't be perfect looking, but it will get the job done, graphically.
Technically? I'm a noob. I've been meaning to look into this so I can fix my own random maps to include some real mountains. I've got nothing yet, but am looking into it.