Had good times in both but would pick Courmayeur of the two.
Cervinia
Generally excellent snow cover (because of the altitude) but can suffer from the wind (altitude!). A good skier might be looking for more excitement after a couple of days as the ski area is not the most varied - but it's very good for intermediates looking for mileage.
Possible to ski over to Zermatt which will keep the experts happy, but the link can be closed if windy.
Scenery is spectacular (Matterhorn et al). Village itself lacks the character of other Italian resorts but is reasonably lively.
Courmayeur
Not a huge ski area but well-connected with a good variety of runs; fun Blues, challenging Reds and a serious Black or two. Some easy freeride between the pistes up top and some good steeps if you go looking. Possibility to do the Italian side descent of Mont Blanc - not done that myself but on my list of must-do runs for the future.
Scenery spectacular (Mont Blanc)!
Attractive village centre with good restos and a chilled-out apres (recall several bars full of lazy sofas and open fires). Bars and ski shops tend to be on the pricy side for Italy but if you have a car you can get out of town (or to Aosta city) for better value.
Very popular at weekends with day-trippers but 99% of them "ski to lunch"; pistes very busy between 11:00 and 12:00 on Sunday and virtually deserted from 12:00 until 15:00 - fantastic time to ski!
Verdict
You'll have a great time in either - if you're doing it DIY your best bet could be a few days in each.