Actually, the songs are only stored in 1 place, but they can be accessed through the different sources.
I think you’re facing the old MSC-MTP USB mode situation.
When your player is connected to your computer under USB MSC mode (set under the player’s Settings), your computer only sees the files that had been transferred to the player when the player was connected under MSC mode. Likewise as to MTP mode. Regardless, all the files show up and are playable on your player, under whatever USB mode setting.
A good reason why some people prefer to set the USB Mode on their player to either MSC or MTP: on the Auto setting, the player tries to connect under MTP mode but then will fall back to MSC mode if the MTP connection does not work; the result being, you may have a mix of MSC and MTP mode transferred files on your player, making some of them not visible on your computer without switching to the other USB mode.
Because MSC mode is more universal (playing with a broader range of operating systems, including Linux and on Macs, and without a need for Windows Media Player on the computer–with MSC mode, the player is seen like an external drive (which it is!)), some people prefer MSC mode. However, note that DRM (digital rights management) files need MTP mode–they will not transfer sufficiently (the DRM license will not transfer) under MSC mode.
Assuming that’s what you’re facing, what you may want to do is: (1) under the player’s settings, set the USB mode to either MSC or MTP mode, and then connect to your computer and transfer all the content your computer then sees back to the computer, and then delete that content from the player; (2) disconnect the player and in its Settings, change the USB mode to the other mode and then repeeat the last step; and (3) then setting the player to the USB mode you want to use consistently, transfer all the content back to the player. At that point, your computer should be able to see all your content on the player at the same time.
Note also: under each of MSC and MTP modes, the icons you see on your computer are slightly different, as a different system of connection is being used. This may explain the difference you are seeing with your new Clip as compared to your old. If you want to be consistent to the old player, just set the USB modes of each player to the same.