Software as a Service (SaaS) is usually hosted in a cloud such as Microsoft Azure or AWS, with support, maintenance and upgrades included in the price. It is normally a multitenant platform: all clients share one instance, and administrators, instructors and learners are separated by the company they are registered under. Companies can usually brand it with their own logo and color scheme. There is nothing to install, which makes it a good option for any organization that wants to implement training quickly with minimum expense. The downside of a single shared instance is that very little customization is allowed and learning objects cannot be shared among several companies.
Platform as a Service (PaaS) is also cloud hosted, but gives much more flexibility. Every customer has their own instance, so branding, customization, integration and content sharing can go much further. Support, maintenance, updates and upgrades are still provided for your instance. That is also why PaaS costs more than SaaS: in SaaS everybody shares hosting, support and maintenance expenses, while in PaaS the instance is yours alone.