Do note that while the table shows that silicone seems to be extremely permeable to CO2, for our purposes, it does not matter (the flow rates of CO2 that we use might be described as "not even flowing" or "an acceptable leak" by some people!)
As for tubing, your regular airline tubing works fine and is cheap, but hardens over time.
Silicone tubing is a little more expensive, but takes a bit longer to harden.
"CO2 tubing" will not harden with time, has a thicker wall, but is the most expensive of these three options.
That particular Clippard tubing doesn't seem to be the correct size. The URT1*** tubing comes in 1/4" OD by 5/32" ID. You probably want the 1/4" by 1/8", which is a slightly different type - poly 85A model # URH1***, and isn't listed on the Cole-Palmer chart, but probably works just fine.
The theoretical correct model number for the URT1 1/4 by 1/8 tubing would be URT1-0804-***-***, which doesn't exist, according to their catalog (you'd have to call to check). The definitely existing 1/4 by 1/8 tubing is URH1-0804-***-***. This, however, is currently out of stock.
The *'s are CLT-050 for clear tubing, 50 feet (the minimum order is 50). And I just realized something - the existing URT1 tubing is URT1-0805, and the 8 is 8/32 (OD) and the 5 is 5/32 (ID). The URTH-0804 is 8/32 by 4/32, or 1/4 by 1/8.
One last thing - Clippard tubing, the URH1, is cheaper at $23 per 50ft (which is their minimum length). Tygon R-3603 would be $30 per 50ft (though you can order any length). Additionally, Clippard charges a *minimum* of $17 S&H on any order (your $3 adapter would cost you $20, for example). I don't remember what the USPlastic shipping is. Anyway, you'd have to do some math to figure out which was more cost effective, based on the length you'd use. I like the Tygon, and have never used the Clippard (except a very old inventory length of tubing, which had clearly gotten much stiffer over time, so that shouldn't count against it), so I'd recommend the Tygon.
The parts number for 50 feet of clear Urethane Clippard tubing in 1/4" OD x 1/8" ID is: URH1-0804-CLT-050 as kevmo911 mentioned. Here's Clippard link. Their new web site is a bit different than their old: http://www.clippard.com/part/URH1-0804-CLT-050