NTFS and ext3/4 and in fact newer filesystems in general support privileges (that is the problem) as well as large files and other goodies -- FAT is old enough that it doesn't support filesizes larger than 4GiB. That is per-file, the actual filesystem
volume can be far larger. My 32GB flashdrive is quite comfortable on FAT32, and depending on the cluster size you can have a max filesystem size ranging from 2TiB to 16TiB.
(technically, FAT32+ can store 256GiB files, but that isn't very well supported and non-FAT32+ drivers will truncate the files. Which defeats the purpose of using a shoddy filesystem format because it is universally supported...
)
tl;dr -- FAT should be perfectly safe.
Why did the NTFS volume work just fine? Because, in addition to using a permissions-supporting filesystem, you also have to remove the permissions from the "Everyone" SID, since I think Windows by default allows everyone rwx except inside an actual userprofile. Linux by default sets restrictive umasks that denies write access to other users -- making an fstab rule for that drive mounting it with a umask of 000 should ensure all new files are created with permissions of 666.
That would mean you don't have to reset the permissions with sudo in the future, and you don't have to use FAT.
Similar discussion:
http://ubuntuforums.org/showthread.php?t=1502775
What is umask:
https://wiki.archlinux.org/index.php/umask