The short answer is that for this length of time I'd choose Nice.
But please remind us of the timing; what month? The farther off-season you are, the more I'd favor Nice because of the potential for the mistral in Aix-en-Provence.
And what's your position on renting a car? Having one for weekend jaunts would be more important in Aix than in Nice.
Nice is about 2.5 times the size of Aix-en-Provence. I spent 2+ weeks in Nice in 2017 (taking French lessons) and just day-tripped to Aix, so I can't make a clear comparison. However, Nice felt a lot larger; I'd have guessed the population difference was a lot greater.
Aix is certainly large enough to provide a variety of restaurants, etc., but it doesn't have a huge number of local sights. On the other hand, it's practically a suburb of Marseille (12 minutes by TGV), a real working city.
I think Nice has a more resort-like feel to it. It has at least six art museums, a lively market and some other sights that will help keep you occupied on days when you don't leave town. It's on the coastal rail line that provides quick access all along the Riviera at a reasonable cost. You'd even be able to pop over to Italy; Ventimiglia's less than an hour away and San Remo takes about 1-1/2 hours with a change. Nice is also a significant bus hub for trips up into the hills, though some of the hill-town destinations require a combination of train + bus.
One thing about the Nice apartment: Be sure the reviews don't indicate infrastructure issues. I think Rick recommends hotels outside that area because the Old Nice buildings are old; I suspect he is concerned about things like plumbing, air conditioning and Wi-Fi, but I don't remember the details.