Yes, it's needed. As you might suspect, the flex pipe is there to flex. The motor moves in relationship to the chassis. The exhaust is connectet to both. It would be bad if there was nothing to flex in it.
The braiding on the flex-pipe is a shield to protect the flexible (think dryer hose) part of the pipe and to a lesser extent disperse heat.
The flex-pipe helps eliminate movement in the exhaust, a major cause of exhaust failure due to breaking welds and leaking gaskets.