The on-premise version of SAP S/4 HANA is a solution that offers maximum control however, it will be installed on your enterprise's servers and will be maintained by your company. Upgrades for this version are released on an annual basis. For the upgraded version to be used, your company's team will have to implement and test this version.
The cloud version of SAP S/4 HANA is a type of Software as a Service (SaaS) that is hosted on SAP's servers and is also maintained by SAP however, since SAP is hosting it on its servers, your company has less control over it as compared to the on-premise version of SAP S/4 HANA. It receives upgrades on a quarterly basis and this specific version is available to enterprises in both public and private editions.