Depending on your level you should be using one type of up-haul or another. In the initial stages, when we rely a lot on the up-haul to maintain our stability on the board when lifting the sail out of the water, we need a rope that is not elastic. If we lift the sail up with and elastic rope, the control over the sail will be greatly reduced as when we pull the sail towards us, some of that energy is lost in the elasticity of the rope.

Beginner Up-Haul

These are a rigid rope that is attached to the boom and has an elastic rope that goes from the center of the up-haul rope to the mastfoot. Since there is a loose piece of rope dangling near the mast base they are not practical for high speeds, especially with choppy water as the loose end will be flapping around and hitting the sail all the time.

Advanced Up-Haul

Once we can do the water-start we are less reliant on the up-haul. We may only need one in circumstances where the wind drops so much that we cannot do a water-start any more. For these cases we use an elastic rope as it will be close to the mast the whole time that we are sailing and not be a nuisance.
The downside to this is that when we lift the sail we don’t have a direct response of the sail. This means that when we start to lift the sail, the rope stretches first and then starts to lift the sail out of the water. If as we are lifting up the sail, a bit of chop hits us, or a small wave, or we just lose our balance a little bit, we have a harder time to stabilise ourselves with the weight of the sail because as we pull, we first stretch the rope instead of the rope being pulled taught right away and helping us catch our balance again.

Summary

Until the water-start use a non-elastic rope as it will provide you with more control over the sail when hauling it up. After learning the water-start you should get an elastic one just in case you need it.