Cloud PMS runs as a multi-tenant or hosted service: you log in over the internet, the vendor patches the platform, and integrations (OTAs, payment gateways, messaging) are usually API-first. On-premise means software and often the database live on hotel or hotel-owned infrastructure you maintain—or pay an IT partner to maintain.
Both can be “real” systems of record. The trade-off is who owns uptime, upgrades, backups, and how fast you can ship GST, channel, or security changes.

