Man if FlashFXP can't do it, I don't know of any that can.
Use:
- Schedule... (load up FlashFXP/connect to FTP server...)
- Load Queue (queue up all files in your web dir)
- When File Exists (it compares file sizes, you can tell it to auto-skip if same)
- On Transfer Complete (quit FlashFXP)
You can probably get it to work if you do some tweaking, but that should be the basis of it. You can also enter custom commands... very robust FTP program.
Alternatively, you can try one of those programs (if they exist - search other threads here) that can download a whole website on multiple levels. It'll just grab every file each time (modi'd or not), but you'd be sleeping while it does it, so why would that matter eh ?