I walked the HRP from Hendaye to Gavarnie last summer (2011) which follows part of the GR10 in France but crosses into Spain on a regular basis. The GR10 goes from village to village ascending into the Pyrenees and then retreating back to another village for overnight accommodation so it is good if you do not want to camp but you are tied to an itinerary which goes from village to village with a lot of ascent and descent.
The western or Basque part of the Pyrenees is generally moist and forested with frequent fog potentially causing some difficulties but full of dark forests, dolmen, stone circles, and other celtic remnants - quite ethereal in many ways.
If you are willing to camp I would highly recommend the HRP (Haute Route de Pyrenees) which forgoes the nationalistic routes either side of the border (GR10 in France and GR11 in Spain) and weaves a far better route by combining the best of the GR10 and 11 without any increase in the daily elevation rise and fall because it tries to stay high and that often means that it is above the valley fogs. I walked for 2 day from Col de Bagargui (Iraty) to Lescun above the valley fog in beautiful sunshine while the GR10 was in perpetual gloom.
You can divide the western third in two - Hendaye to Lescun which is the Basque country and then through to Gavarnie for some superb high mountain walking. I am returning in 2013 to complete the HRP through to the Mediterranean.
There is a Cicerone guide to the route by Toon Joosten as well as the original guide (in French) by Georges Veron. I recommend the Joosten guide as the route is better thought out if you are happy to camp as the Veron route is based on accommodation each night.
Walking my own walk