You usually want stretchy if using a vehicle to pull another out, the idea being to multiply your vehicle's pulling capability by converting mass x speed into kinetic energy to unstick the stuck vehicle. If you just pulled slowly forward until a non-elastic rope or chain got tight and then tried to pull, you would need to put more horsepower on the road and could easily lose traction before breaking the hold on whatever the stuck vehicle is hung-up on.

Non-stretchy is for use with a winch or come along attached to an immovable anchor (tree, boulder). Amsteel rope is sold to replace steel aircraft cable on recovery winches, so it doesn't stretch much.

I always carry a 30' of 4" web snatch strap (stretchy), 12' of non-stretch 3" web strap (tree strap), 12' of heavy chain. In the winter or when off-roading I add either a 2-ton come-along or a Hi-Lift jack, or sometimes both. I've used the snatch strap several times, and the chain, tree strap and come-along a few times, to get others unstuck. I've been lucky enough to never get seriously stuck myself.