'Salesforce for Outlook' icon in system tray disappears after computer is restarted

The icons for installed applications are hidden in the Windows Notification area and you would need to manually unhide them. When you have hidden icons an arrow appears next to the Notification Area at the bottom right corner.

On Windows 10, see all your notification area icons by clicking the up arrow to the left of your notification area icons. Or, check Microsoft help for Windows 7.

For Windows 8 or 10, Microsoft doesn't have a similar link but there are a few great 3rd party sources that explain the details with screenshots.

Show SFO icon on the taskbar

1. Right-click an empty area on the Taskbar, and then click Properties.
2. Under Notification area, click Customize.
3. Select the 'Salesforce for Outlook' from the list of Icons and select 'Show icons and notifications' from the drop down on the right under Behaviors and then click OK.
4. Your SFO icon should show up in the System tray now, even if you restart your computer