I would say a mix of both.
crunches work a different set uof muscles then sit ups do. So i do both in my workout. i am also a huge fan of leg lifts to work on the lower tummy. But you just have to make sure you do them right. Like sit ups make sure you press your lower back into the floor and use your abs not your arms or legs. And you can do them slowly, that actually makes them more efficient. Slowly might also help with the pain.
but i have to play devils advocate..i agree with dh
sit ups invlve crunches and add a little extra, so technically i would say they are better...