The saddle on the left is the basic B.17 Standard. Most modern bikes have a micro-adjust seat post that has the clamp built in and, in that case, the basic B.17 is all you need.
Some older bikes have a plain-top seat post that requires the use of the clamp.
For some pictures of the two types of seat post and the clamp in action, see Mounting a Double Railed Brooks Saddle . In the Brooks line only two single rail models come with a clamp included, the B.17 N and the black B.17 Standard. Clamps are included with all the double and triple railed models of Brooks saddles.