Android 4.3 update for Samsung Galaxy S4 Google Play edition prevents apps from writing to microSD card18
In the meantime, you can go to the Google Play Store and install ES File Explorer which happens to be in the right group for writing to and reading from a microSD card. If your phone is rooted, you can get into the files and make the appropriate fix yourself directly on the phone, or from your PC. In basic terms, you are telling your phone that it is ok to give permission to the media-rw group to write to the microSD card. Once you get into the /system/etc/permissions/platform.xml file, you need to focus on line 71. Check the photo below. All you are doing is adding that permission.
If you don't want to root your phone or you're worried about bricking it, you can always stick to ES File Explorer. And hopefully, a fix will be sent out soon.