My two cents from my experience:

QR:
+ Better Food
+ Doha less prone to weather delays

CX:
+ Better seats on longhaul only (QR has 2-2-2 on the 777 with no privacy, 787 and 350 seats are similar but have a fixed table which is in the way when you try to sleep)
+ Stopover - HKG is far more fun than Doha

Wash:
- Service usually good on both
- Lounge (Doha has more food options and a sit down restaurant, but can be far to gate)

Both are good airlines. Price, schedule and airplane type would be my deciding criteria (avoid QR 777 when traveling alone, avoid CX regional for longer flights)