The reason that one end is fixed is for the piping design. In piping design, it is necessary to provide locations where the displacement is zero - directional anchors and guides. This is necessary so that the thermal growth of the piping can be appropriately managed. Having a location of zero thermal expansion in a horizontal vessel is therefore necessary to accurately predict the thermal displacement of nozzles, so that the attached piping doesn't over-stress the nozzles.
If both saddles were sliding, then the thermal displacement of the nozzles would be unknown/indeterminate.
Also, if both were sliding, then the vessel could "walk" over time. If you could imagine that upon first start-up, the left saddle is "fixed" and the right saddles slides, but upon cool-down, the right saddle "sticks" and the left saddle slides. Repeat. When this happens. vessels can translate large enough distances so as to fall off concrete pedestals, etc, causing substantial damage. If the piping is small, then it just gets dragged along for the ride, usually ending up a bent mess. If the piping is large enough, then nozzles could be over-stressed/ripped out.
Make sense?