I've ridden both roads a couple of times, in both directions. I like Route Napoleon and love the Route des Grandes Alpes. But I do like mountain riding. And will put up with the occasional stretch of poor tarmac. There are a couple of things worth adding to what's already been said:
Which order? Personally, I'd go south on Route Napoleon and then north on the RdGA. Then you're building up, from the broad, sweeping Napoleon to the higher, tighter, more challenging passes. That way, each day can get better than the one before.
Two questions with RdGA. First, which version do you take? The "classic" version of the RdGA - heading north to south - crosses Col de Vars and then heads through Barcelonette to take Col de la Cayolle and Col de la Couillole. A lot of riders prefer to cross Col de Vars, get as far a Jausiers and then head south over the Col de Restefond and Col de la Bonette (taking the loop above the pass to the 2802m Cime de la Bonette, France's highest paved road and the highest non-dead-end paved road in Europe). Both are good, but La Bonette has a better surface, bigger views and is shorter/saves a couple of hours of demanding riding. To be honest, if it's your first time doing the route, that's the way I'd go. Unless you want to tick every box, in which case accept that Couillole is pretty damn rough and Cayolle ain't exactly smooth (but see below on the subject of detours).
Second, do you need to do all the route? For me, the north end of the route from Thonons les Bains down to St Jean de Sixt is lovely but missable: nice roads, but you'll ride better if you look for them. From St Jean you pick up Col d'Aravis, which is the first truly great pass. That's followed by absolute magic roads: Cormet Roselend, Col de l'Iseran, Telegraphe, Galbier, (my favourite) Izoard, Vars... and your choice of La Bonette or Cayolle. But then you get to the southern end dropping into Menton over Col de Castillon is a bit anti-climactic after the previous highlights; while half the Col du Turini is lovely, the other half has an atrocious surface (or did this time last year); and St Martin… well, you're only riding that one to get to the Turini. I'd be quite happy riding between St Jean de Sixt and the D6202 (though, okay, I do like the town of Sospel at the southern end of the Turini).
Finally, detours. There are a couple of places worth trying to spend two nights in, to do some extra riding. Top of my list would be Castellane, at the end of Route Napoleon (okay, it goes further, but that's more or less the end of the great riding - there's a few good miles after that, then you're into Riviera traffic.) From there the loop around the Gorges de Verdon is one of the greatest rides in France - absolutely on a par with Route Napoleon and the RdGA. Build a day in for that if you can. As has been wisely suggested, stopping in Barcelonette to do a bit of extra riding is also well worth the effort. That'd allow you do ride up the RdGA on Cayolle, but then do La Bonette and loop round on the D6202, up the Gorge du Cians, down the Gorge de Daluis and over the Col d'Alos - another belting day's riding. Likewise, stopping in Bourg St Maurice to ride the Petit St Bernard, Grand St Bernard, Forclasz and Cormet Roselend (just make sure you set the route so you ride that one in the opposite direction to the one you'll take for the RdGA ride) is highly recommended.
Of course, all this adds time, cost, inconvenience, etc etc - but it would be so very worth it.
Edit to fix autocorrect changing place names