Both the Anti Roll Bar and Torsional Bar / Torsion beam work on the same principle - torsion or twisting of a beam to provide an elastic mechanical connection between opposite sides of a suspension.
I've differentiated between Anti Roll Bars and Torsion Beam Axles in the following sections
Anti-Roll Bar ( henceforth referred to as ARB )
As the name suggests, an ARB prevents excessive sideways tilting movement (Roll) of the vehicle while taking turns. The anti roll bar is usually a thin and shaped bar that connects two sides of an independent suspension system ( mostly used in the front wheels ) so that when one suspension moves up or down, the movement is PARTIALLY transferred to the other wheel too and thus helps maintain a mechanical connection with flexibility ( due to torsion ).
An ARB is also referred to as a Torsion bar sometimes. The ARB is attached by means ofpin joints or ball joints to the suspension linkages on either side and by means of bracket mounted pin-joints to the under body of the vehicle.
These linkages are a key difference between Anti Roll Bars and Torsion Beams. Torsion Beams are typically welded to the trailing arms of rear axles
Couple Torsion Beam Axle / Twist Beam Axle (henceforth referred to as CTBA )
A CTBA is a much thicker formed metal beam that is welded to a set of trailing arms. The torsion beam acts as a flexible mechanical link ( similar to the anti roll bar ) with a high torsional stiffness compared to the anti-roll bar
Since the CTBA is welded at both ends , it allows for a significantly lesser range of relative movement between the two trailing arms. CTBAs are used in rear suspension in tandem with coil spring and damper systems in small cars and other entry level sedans.