Wood generally looks better and weighs more and weights not necessarily a bad thing on a spring gun. Sure, it's harder to carry, but it tends to make the rifle hold steadier and make it less hold sensitive. On a spring gun the recoil comes from the piston slamming forward to compress the air and that means you experience the recoil before the pellet leaves the barrel, so anything that's different about the way you hold the gun will cause it to shoot to a slightly different point of impact. This is called hold sensitivity. A very powerful and lightweight airgun will tend to be very hold sensitive, while with a low powered and heavy airgun you might not notice any difference at all. A hold sensitive gun will be more difficult to shoot accurately. The gun is often capable of good accuracy, but holding it in just the right way every time makes it harder to achieve.
Plastic and wood both come in very different grades. Really expensive wood stocks tend to be both beautiful and elaborate, but not necessarily any more functional than a cheaper wood stock. Really cheap wood stocks tend to be made out of ugly wood and covered in shellac to make them look darker and if you strip that off you might find filler underneath. That's what you might find on a low end Chinese gun. High end synthetic stocks often have rails or tactical features built in and often come with camouflage finishes. They feel as solid as a wood stock and usually weigh as much and they are much less affected by heat or humidity than wood. Really cheap synthetic stocks tend to be very lightweight and feel like the cheap plastic they are. For a low end gun I'd prefer wood over plastic. There's also a vast middle ground of ordinary stocks that aren't necessarily ugly or made of cheap plastic.