There are three possible conclusions that one can draw from your question:
(1) You have some sort of physical limitation which causes the discomfort,
(2) you are not executing it correctly, (incorrect execution can cause discomfort), or
(3) you're still fairly new to the concept. Over time, repetition and conditioning will make it easier and more natural for you.
Because I have not worked with you in person, it would be impossible for me to guess which of the above applies to you, and to what extent. It could be any one of them, or a combination of some or all. It is true that it is most likely some combination of #2 and #3, but it would be dangerous for me to rule out #1 without ever having seen you. That is your teacher's job.
But, just for the sake of discussion, I will assume that #1 does not apply. Let's look at #2:
With quick head snaps, proper technique is ipmerative not only for your safety, but for speed as well. You have to apply the right amount of muscular tension and at the right time. It could be that you simply apply too much tension all around, which actually serves to slow down the movement. There is a brief moment when maximum muscular tension is appropriate, but if you sustain that tension throughout the entire movement, it doesn't help (and could even hurt!). I can only go so far to describe it here... the rest is up to your teacher. The most important thing for you to understand is that there is an actual technique; It's not just brute force.
Now it could also be that your teacher has already explained it well, and your execution is not bad. In that case, we'd have to look at #3: Conditioning. Sometimes it just takes time and experience to get used to something. Actually, even if you do think you have the technique covered, experience just continues to refine your technique, with or without additional instruction, and after a year you'll notice that it's not nearly as bad as you thought it was in the beginning.
But here's where you have to be careful: Although there's a good chance that with proper guidance you will continue to improve, you can't absolutely assume that things will just naturally get better. They probably will, but you're doing something funky that nobody is helping to fix, that too will embed itself into your form. So it's possible that things won't just naturally get better.
The good news is, if you do have a reliable teacher or coach who is good at fixing these sorts of "anomolies", you can rest assured that time and practice will get you past this.
Good luck!
Sincerely,
Jonathan Atkinson
www.ballroomdancers.com