Both the certificate of incorporation and registration are different in terms of company and LLP. In this case when the company or LLP is registered with Ministry of Corporate Affairs, registry issues the Certificate of incorporation. It will be used as proof for incorporation and certificate will include the name and date of incorporation of the company or LLP. Further when such company or LLP takes registration under any other ACT such as GST or IEC or Shop & Establishment, etc. Such certificates are known as certificate of registration.
In case of proprietorship firm, there is no specific registration for incorporation. It takes registrations under different ACT such as MSME and other as mention above. In such cases both the certificate of incorporation and registration will be same.