I have been doing an exercise DVD and last night did the abs section for the first time which involves a fair few sit-ups. At the time, I was aware of my neck being rather uncomfortable and this morning it feels really stiff. My abs don't ache at all - just my neck
Did you habve your hands behind your neck? That is wrong.
If you have very little core strength (i.e no ab muscles) then your back and neck would have been doing all the work to lift your torso off the florr, but as you gain core strength you should be able to do them better.
Basically if you start sitting up and hugging your knees, then gently roll your torse back in a controlled way, you should feel it in your abs.
Good sit ups shouldn't have you lunging forward right up to your knees and back down again, but rather they should be slow controlled movements where you lift your torse off the floorand then move up and down keeping your stomach muscles contracted. You don't have to sit right up and down again each time, 'crunches' are best.
Heath I don't think you can get a sticky out abdoment from doing ab workouts
It is absolutely ok for your hands to be behind your head, but your neck should be relaxed (not tense) and head supported by your hands, not pulling your head up. They are there precisely so that you can avoid working/straining your neck but it can take practice to properly relax - don't tuck your chin into chest, keep head in a neutral position, in line with spine. As mydeadbody said, there is also no need to sit all the way up.
You can get sticky out abs from too many crunches and sit ups, especially with poor technique! Better to do more planks, side planks and Pilates type exercises as well as some (rather than loads) of crunches. Also don't forget your lower back which is really important for core strength and injury prevention - try adding some supermans to your routine.