Pro gives full features like remote printing, bandwidth monitoring, and a little more control over screen sharing resolutions and speeds (HD can be used here, but why would you want to?)
Rescue is used for ad hoc support (like teamviewer).
Central allows for central control over all your accounts (free and pay) and comes with 100 free logmein accounts.
So with Central I create a deployment package that I then install while I am on site OR send them the install link, they install, and I can manage their systems remotely.
You can also delegate access, so I actually send them a shared account so they can access their own work systems remotely from home.
So with Central I will install (for example) 5 computers for Company X. I group those 5 systems together, and then can grant the company owner access to the account (via email invite).
I track their logins and ALSO retain access to their systems.
I usually bill them for the central access at the same rate I pay, so it is a wash as far as costs are concerned... but the gain is they get remote access and I can do upgrades/maintenance after hours. My clients WANT to pay for access, but usually don't want the hassle of managing the systems etc.
Central comes with a nice desktop client (used to be called ignition) which allows for quick access to all your desktops (saving tons of time).
