Can't write to my FAT32 USB drive on Mac
I have seen that issue (disk being read-only on Mac and read-write on Windows) before where it was caused by a slight disk corruption.

I solved it by plugging the USB drive into a Windows PC and running scandisk to check for and correct errors on the drive. Afterwards the drive worked just fine on both Mac and Windows for reading and writing.

Note that it is important to "safely eject" the drive whenever you want to disconnect the drive from the Mac or PC. On the Mac, you'll need to press the small up-arrow besides the drive name in Finder (called unmounting). On the PC you'll need to look in the dock for the icon for safe ejecting hardware, select your drive and eject it. Another option is to power down the PC/Mac whenever you want to unplug the drive, but that is cumbersome.

If you do not safely eject the drive, you risk corrupting the drive.