Private Cloud Hosted: Run SAP Business One SQL or HANA version. Best for situations where your solution requires add-on products to SAP Business One.
Public Cloud Hosted: Run SAP Business One SQL or HANA version. Good for situations where you have minimal customizations, and can run the standard offering.
SAP Public Cloud Hosted SaaS: Run SAP Business One HANA version only. Best for situations where you can run the standard offering, and prefer a subscription pricing model.