UPDATE: If you are running the latest versions of Windows Live Essentials (15.4.3502.0922 or 15.4.3508.1109), then here is how to get the minimize to System Tray functionality back.
Modify the properties for the shortcut as shown here, then once you have Live Mail open right click on the System Tray Icon and make certain that “Hide window when minimized” is selected. Afterwards the Live Mail app window will minimize to the System Tray just like before.
Windows Live Mail in Windows Vista
In Windows Vista you could minimize Windows Live Mail to the system tray if desired using the context menu…
Windows Live Mail in Windows 7
In Windows 7 you can minimize the app window but not hide it in the system tray. The Hide window when minimized menu entry is missing from the context menu and all you have is the window icon taking up space in your taskbar.
How to Add the Context Menu Entry Back
Right click on the program shortcut(s) and select properties.
When the properties window opens click on the compatibility tab and enable the Run this program in compatibility mode for setting. Choose Windows Vista (Service Pack 2) from the drop-down menu and click OK.
Once you have restarted Windows Live Mail you will have access to the Hide window when minimized menu entry again.
And just like that your taskbar is clear again when Windows Live Mail is minimized.
If you have wanted the ability to minimize Windows Live Mail to the system tray in Windows 7 then this little tweak will fix the problem.