A snaffle in technical jargon actually only refers to the mouthpiece, but colloquially is often used to refer to the complete bridle. The term "snaffle" is therefore used for the complete headpiece of the horse, which consists of the browband, neckpiece and cheekpieces, the bridle, i.e. the bit, throatlatch, browband and curb strap. Here a bridle with a wide noseband completes the set and all these pieces together form the snaffle.
The bridle itself can therefore be changed by different variations of the actual noseband, but the snaffle with its different parts is also absolutely necessary in order to be able to use the noseband at all.