I understand that Plan-G sources its maps from OpenStreetMaps. If, however, I fly repeatedly in the same parts of the world do I need to keep connecting to the internet or should I presume that the maps are cached first time and there is no need for them to be downloaded again and thus no need for an internet connection every time?

Hold the Horses! I've just launched Plan-G again and it's working alright now. I have no idea why!

I have been using Plan-G3(v3.0.2.51) on a laptop running Windows 7 64-bit without problems.
I have installed the same software onto a newer laptop running Windows 10  but the maps do not appear. I know that it is working with FSX because the little yellow aeroplane icon orients correctly and leaves a trace as it flies around and also the nearby navaids and aerodromes are displayed.
However, the underlying map is not there, just a lot of blue tiles with the wording "Exception: the remote server returned an error: (403) Forbidden".  The internet is active.
Be sure that the map source is set to OpenStreetMaps and I can access that website normally in my browser.
Strangely it all works OK on the Windows 7 laptop. Any help in fixing this problem would be much appreciated.