Personally, if you have a free choice, I do not see any advantage of hiking south to north. However, you may have personal or business reasons to be on the French Riviera at the beginning of your hiking period, or to be in Switzerland or nearby at the end of your trip.
If your thought is to avoid snow in the passes by starting in the south, this is unlikely to be the case. The highest pass on the entire GR5 system is the Pas de la Cavalle, at 2,671 meters (8763 feet), and that is in the Mercantour Park, a walking week or so from the Mediterranean. The Pas de la Cavalle is about 150 meters (500 feet) higher than any of the northern passes on the GR5.
Weather, in my opinion, should not be a reason to hike northward. In the more-arid south (but not near the Mediterranean) daytime highs will be higher than further north (but drier), and nighttime temperatures will be cool. If you begin hiking at 8:00 am (which is about 6:30 am sun-time) you will be across the passes and hiking downhill by the later part of the day. However, if you hike northward, the long climbs of the first few days will be in a hot-humid Mediterranean climate.
A difficulty of hiking south to north is that the guidebooks mainly emphasize the opposite direction.
An advantage of ending in the south is the ability to celebrate with a day at the beach.