There isn't much caffeine in a diet Coke. I think it is around
30mg/can.

I do know many people complain of headache with aspartame.
That is the most common thing I hear about it.

Diet Coke is addicting to some. We have a cousin who consumes tons of it. Also I had a chronic pain patient who consumed several 2 liter bottles a day!

There is a caffeine withdrawal headache, but you'd have to consume alot of Diet Coke to get into that range. Caffeine withdrawal happens more commonly with coffee drinkers.
I've had minor type symptoms from tea withdrawal at times. (2 strong cups/day can affect me).

Drinking alot of cola type sodas gives people too much phosphate. This can affect calcium levels in some people.
Some believe that this can affect bone metabolism. However, there are not typically noticeable symptoms from elevated phosphates.