1) If you wait too long before seeking care, the insurance company for the responsible party will argue that you aren't really injured or you wouldn't have waited so long.
2) Seeking medical attention right away is strong evidence that you were actually injured.
3) Going through your own health care channels, at least initially, is often persuasive that you are injured.
4) Seeking medical attention early helps document your injuries. Making sure you mention ALL your injuries during your first appointment is persuasive to an insurance company. Contrast that with NOT mentioning all your injuries and trying to introduce a new injury a few weeks later. That certainly looks suspicious.
5) Finally, seeking medical attention is more important than who is going to fix your car, or who said what to who at the accident scene. Don't get stuck on issues that don't matter as much as your health care. Keep your "eye on the ball." Your health is arguably one of your most important assets. It's simply a wise thing to do regardless of whether or not you choose to bring a claim against anyone.
If you've been injured, you should also seek medical attention right away. It may seem obvious but many folks wait too long or don't get any care at all.