There is a difference between Google Play Music which is located on Googles servers and appears on all android devices you are logged into Google on ( your Gmail Account).
And the Music player on the device. If you uploaded to Google Play the music is there. You just have to make sure you are syncing to Google in order to listen to it.
Now a SD card although great for storing music and photos often times is not formatted for that device. It should function from phone to phone and you did say you can see the files via computer? Via file manager on the phone?
What I would do is transfer all the music on the SD card to a folder on your computers desktop. Then insert the SD card into the phone and format it. Then hook the phone back into your computer and transfer the music and photos etc. Back on the SD card in the device.
Open the Music player and see if you can play the music. If you can the problem is fixed. If you cannot it could be a bad phone, or a bad SD card.
However as I said above, if using Google Play Music your uploaded songs using the service will remain on their servers.
To see if the music on Google play is still there. Log on through the web interface from your computer and verify the music is there.
I have found that Google Play Music is getting more and more restrictive in an attempt to become more iTunes like, which in my mind is NOT a good thing. The last time I tried using Google Play Music and wanted to have the music files located on my phone, Google Play Music wanted these files within the app storage, not simply listed in the file explorer structure as separate files. Not sure this is still the case, but it was then. This is the reason I have stopped using the app. There are a multitude of reasons why it is better to have your music on your phone rather than in the cloud and playing from there. Having an increase in data usage is but one of those reasons and by no means the biggest reason.
You may try opening one of your music files while in your file explorer. When the dialogue opens up in order for you to choose the app which will open the file, if you would like to use Google Play Music, simply choose that app and choose it as the default, or check "always" as the option. Possibly that will get Google Play Music to see the remainder of your files on the SD card.
Another thing you may try is to use a different music player. I prefer using Doubletwist.
Try looking in the settings of Google Play Music, too. Is there a setting for the app to have music storage in internal storage or SD storage. Some apps will make you choose between the 2 options. I would assume that the default would be internal as not all phones have an SD card option.