A quick look shows that both forms ask the voter to attest they are US citizens and warns them of penalties for filling out the form fraudulently, but using the State form requires the voter to show proof, while using the Federal form does not require the voter to show that proof. They can simply show some ID and proof of residence, which doesn’t necessarily prove citizenship.
So if voting a fundamental enough right that you hold it inherently? If you move to a different district only to find that your dog ate your long form birth certificate, should you be disenfranchised until you get a new copy?