I can't write to my flash drive, and it is already in the MS-DOS (FAT) Format. Why?
So, here's what I did after much face-palming and cursing of Apple and their absolute disregard for their users:

From the terminal:  


Identify your USB by NAME and IDENTIFIER:




diskutil list


Output is:

/dev/disk3 (external, physical):  
#:                       TYPE NAME                    SIZE    IDENTIFIER  
0:               FDisk_partition_scheme              *2.0 GB     disk3
1:                 DOS_FAT_32 MIXTAPE                 2.0 GB   disk3s1


In this case, NAME=MIXTAPE and the IDENTIFIER=/dev/disk3s1


Now unmount the USB:




sudo diskutil unmount /dev/$IDENTIFIER


Example:

sudo diskutil unmount /dev/disk3s1


Output is:  

Volume MIXTAPE on disk3s1 unmounted



Now create the Volume directory - this appears to be the key!




sudo mkdir /Volumes/$NAME


Example:

sudo mkdir /Volumes/MIXTAPE


No output.


Now mount the USB to the Volume:




sudo mount -w -t msdos /dev/disk3s1 /Volumes/$NAME


Example:

sudo mount -w -t msdos /dev/disk3s1 /Volumes/MIXTAPE


No output.


Validate that the USB is now writeable:




touch /Volumes/$NAME/tmp.txt


Example:

touch /Volumes/MIXTAPE/tmp.txt



You should now be able to see that you were able to create the tmp.txt file on your USB in the Finder app or by:




ls -al /Volumes/$NAME


Example:

ls -al /Volumes/MIXTAPE