According to Apple Support, "Manually manage" is a means to use your iDevice with multiple computers. From this article:
When your device is set to automatic mode, iTunes automatically
updates your device's music library whenever you connect it to your
computer. iTunes transfers new songs you've added and erases songs
you've removed. The first computer you connect your device to is its
"home" computer. The music library from that computer is copied to
your device. When you connect your device to a different computer, an
alert box appears with this message:
iPod is synced with another iTunes library dialogue
If you want to delete the music library on your device and make the
new computer its "home" computer, choose Erase and Sync. iTunes will
delete all songs and playlists on your device, and then will copy the
music library and playlists from the new home computer to your device.
If you want to move the iTunes Store purchases on your device to the
new computer, choose "Transfer Purchases." The new computer must be
set up with your iTunes Store account and authorized to play your
iTunes Store purchases in order to transfer your iTunes Store
If you want to keep the music library on your device, but copy songs
or playlists from the music library on the second computer, click
Cancel when the dialog box appears, and then set your device to manual
The article also has this caveat:
iPod shuffle and iPhone are intended for use with a single computer.
You cannot load music from multiple computers or iTunes libraries onto
iPod shuffle and iPhone like you can with other devices.
On the other hand, "Sync only checked songs" provides you with control over which songs you want to sync with iTunes - you can select the following items to sync music from: playlists, artists, genres or albums. More details can be found in this Apple Support article.
If you're not using either an iPod shuffle or iPhone, you can combine both options for maximum control over music syncing options.