With the drive mounted, run
and look for your flash drive in the output.That'll tell you what mount options have actually been used and whether the drive has been mounted read only (ro) or not.
Outside of the desktop's automounter, mounts are normally performed by root and default permissions only allow root to write to the mounted drive.
The default driver for ntfs only provides read access, it does not provide write access. If your drive is NTFS, try installing ntfs-3g:
No version of fat supports linux/raspbian file ownership and permissions so these are faked by the driver. Try adding "umask=0000" to the mount options to override the defaults and give full permissions to everyone. This may work for ntfs too. This approach isn't secure but is easy.
Take a look at the man page for mount for more detailed information: