A smaller diameter rope of the same material and construction will bend/mate around the host full size line better. This gives better grip and less strength loss to the smaller diameter 'cord'. Bending a line around itself may work for grip if line is flexible enough; but the strength loss will be much greater than a smaller diameter cord bent around the 1/2" etc. (rope) device.
i experimented for awhile with a 5/8" 3 strand as a split tail for a Blake's around 1/2" lifeline as 'host'. It can be done, but not recommended; the smaller diameter seats and grabs so much surer and quicker.
i prefer 3/8" Tenex around a 1/2" lifeline; it flattens to not grab as intensely on single narrow line like round cord; and in flattening it grips better and with less strength loss.
Real stiff cord, even of the same diameter might not grip as well, and surer; higher strength loss.
The friction hitch / prusiks that gives support from 2 legs of line from friction hitch gives better grab and better release/ redress as Dan-o says. There is less tension per line of support; to grab at least as well and free up easier. Also, the 2 (rather than 1 leg Tautline / Blake's as single leg of support friction hitches) legs of support in the prusiks by halving/ sharing tension between 2 legs in the prussik allows the smaller diameter (therefore less tensile strength generally) to be safe as a single leg of support in larger diameter/ tensile.
These prusiks are for experienced peoples ascending in 1:1 SRT; but not descending. Pros can Ascend/ Descend in the standard 2:1 support of a tree climber's DdRT safely because the bodyweight can shift over to leg of line terminating at saddle and allow the other leg of support to 'stretch' ie. free up friction hitch to slide easily. In SRT there is no other line of body support to switch climber as load over on to. More static line can be used in SRT, than DdRT for same shock absorbing capability; due to elasticity % is different / higher for more weight on the single line; than each leg of the DdRT.
Sorry:taped:
LOts of knots, knot research and science; friction hitches etc