The WebApp
Check out What is the Ideamapper WebApp? for further information on its advantages and limitations.
The Ideamapper desktop offline version (such as Ideamapper Pro) does not access the cloud and is a standard downloadable version with an installer for Windows, Mac and Linux. Your license is attached to the device itself, as opposed to you as a user.
The Ideamapper Cloud for desktop is also an installed version, with installers for Windows, Mac and Linux. However, this version accesses the Cloud, so requires you to login with a username and password, or SSO provider. The license is linked to your login credentials, meaning you can also access the webapp version of Ideamapper.