You may use a certificate of citizenship or a naturalization certificate, to prove that you are a US citizen. But both the documents are not the same. A US citizenship certificate is granted to a person who acquires or derives citizenship from his US citizen parents. But a certificate of naturalization is granted to a person who becomes a citizen of America through naturalization. USCIS issues naturalization certificates to foreign nationals who become US citizens through the naturalization process, after the age of 18. But a certificate of citizenship is issued by the USCIS to an individual who was born out of the country to US citizen parents or parent. A person who was born outside the country may obtain this certificate and use it to prove that he is a US citizen. He may not be required to appear for the naturalization tests, to get a certificate of citizenship. If you feel that you are eligible for a US citizenship certificate, you need to file the USCIS Form N-600, Application for Certificate of Citizenship.
Through the legal process, naturalization, a foreign national can obtain US citizenship. Form N-400 must be filed by the person who wants to become a US citizen. Before that, the person who seeks to become a US citizen must be a Green Card holder. He will be considered eligible for US citizenship only if he satisfies some eligibility requirements. The permanent resident who wishes to become a US citizen must file the application for naturalization and then he must appear for an interview. The interview is considered to be more important as the Green Card holder will have to take the English and the civics tests. After the application is approved and if the applicant is found to be eligible for US citizenship, he will be required to take the Oath of Allegiance to America, at the naturalization ceremony. Then he will be granted a naturalization certificate and he may use that document to prove that he is a citizen of the country and he may not be required to hold the Green Card after that. Remember that only the naturalized US citizens will be granted naturalization certificates.
Holders of both the certificates will be granted US citizenship rights including the right to vote and they can also obtain US passports. Both the certificates serve a single purpose but the eligibility requirements to obtain the certificates are not the same. You need to file different forms to obtain these certificates. A person who is eligible for a certificate of citizenship need not apply for naturalization and get a certificate of naturalization. But you may use the USCIS Form N-565, Application for Replacement Naturalization/Citizenship Document, to replace both the certificates. You may file Form N-565, to replace your naturalization certificate or your citizenship certificate, if you had lost it or if it was destroyed.