The access to a VFAT partition often gets implicitly set to read-only when there are access/read errors. In order to check and solve these try the following steps. Be aware to use the right device, else you can damage your system!
- Try to save valuable data from the drive by copying it to some other drive or your home directory.
 - Unmount the drive.
 -  Check for problems with 
sudo fsck.vfat /dev/... -  If 
fsckreports errors, try to repair them:sudo fsck.vfat -a /dev/....
For more thorough checks see also the-tand-Voptions, as well as-fto try saving the content of damaged files. -  After 
fsckhas finished you can remove and reconnect the drive and try again to access it. -  If that doesn't help, try to format the drive: 
sudo mkfs.vfat /dev/.... - If that still doesn't help, or the problem reoccurs, the drive may be damaged badly and should be discarded.