You might want to look at the "neck pain" portion of this page: How to deal with common cycling ailments - BikeRadar. There is a stretching/strengthening exercise for deep neck flexors recommended there that might help. You may want to see a physiotherapist or similar professional for some assessment and advice.

How large is the saddle to handlebar drop on your bike? If it puts you in a fairly aggressive race position with the handlebars fairly low, that could be an issue. You may want to think about having the stem flipped over so that the handlebars are a bit higher up. This might make using the tops even more comfortable and allow you to start using the drops. Once you get used to riding more you might be able to flip the stem again to lower the handlebars.