There are several documents about interfacing between Arduinos and USB flash drives.
Do I have to use "VDIP1" only, or is it OK to use FTDI USB-to-serial converter?
It seems both methods can use the same FTDI chip.
The VDIP1 is a packaging of FTDI's Vinculum embedded USB host, which gives a simple microcontroller some ability to interact with USB peripherals.
More common usb-serial converters from FTDI and others are USB-device interfaces, which have to be connected to a computer which has a USB host interface, and provide a legacy asynchronous serial interface for connecting to simple peripherals or small microcontrollers. Because they contain a USB device interface rather than a USB host interface, they cannot be used to host USB peripheral such as a flash drive.
A number of modern microcontrollers have built in USB capability. Typically this is device-only in the lower end models, but sometimes the high end models have an interface that can be configured to be either a USB host or a USB device.