MAX232 (3223...) is level shifter, it just takes few 0-5V or 0-3V inputs and shifts the level for them to -9V - +9V .... it also takes few lines with levels of -9 - +9 and shifts them to 0-5V (0-3V)...
The -9 - +9 V side is RS232 and goes to the DB9, the 0-5 (0-3)V side is for ft232, microcontroller etc ... if you need to communicate with PC then you need to shift level to RS232 (-9, +9) but if you talk between two chips, no need to shift the levels to -+9 and then shift them back to 0,5(3) ... just tie them directly... now, it might be correct to call max232 transceiver (my english is not that good so cannot bet on it) but what it does is level shifting.
ft232 is cool chip although i prefere the usb to parallel versions over usb to serial .. for testing **broken link removed** are great