EOSIO and EOS both are part of the same ecosystem, but have different functionalities:
Developed by Block.one, EOSIO is a leading open-source blockchain platform that enables cryptographically secured transaction-layer infrastructure through programmable smart contracts. EOSIO is designed to ensure account security, custody, permissions, and provides a malleable framework to accommodate evolving regulatory guidance.
By comparison, the EOS Public Network is an independent blockchain network that operates on Delegated Proof-of-Stake, built on the open-source software framework of EOSIO. The EOS Public Network is one of many open networks that use EOSIO as a foundation. The EOS Public Network empowers people and organizations to create and access a wide range of decentralized applications and ecosystems. It is not controlled by any single entity, but guided by a cooperative community of digital asset holders.