PowerShellGet is the Package Manager Manager for PowerShell modules.
PowerShellGet is the package manager for PowerShell items. There is a PSModule package provider that exposes PowerShellGet functionality through PackageManagement (aka OneGet). Because of this, you can either run Install-Module or Install-Package -Provider PSModule to install a module from the PowerShell Gallery. Certain PowerShellGet functionality, including Update-Module and Publish-Module, cannot be accessed through PackageManagement commands.
In summary, PowerShellGet is solely focused on having a premium package management experience for PowerShell content. PackageManagement is focused on exposing all package management experiences through one general set of tools.
PowerShellGet can install modules from PowerShell Gallery, NuGet feed and any other source supported by OneGet.