There are a few properties that a wooden spoon has that make it favored by chefs:
It’s strong; this means that it can stir thick things without breaking, and it can scrape things off of the bottom of pans;
It’s soft; this means that, with the possible exception of nonstick pans, it’s not going to scratch the finish of your cookware;
It’s insulated; this means that, if you’re making candy or something else sensitive to sudden temperature changes, you’re not going to cause sudden crystallization by putting in a tool that will suck a bunch of heat out of part of your dish;
It has a high heat tolerance; this means that it won’t melt if it rests on the side of a pan or stirs something really, really hot;
It’s wood; this means that it looks nice, and also that it feels nice in the hand. For those who went through classical training and interned with someone also classically trained, there’s a good chance that they learned to cook with some wooden spoons, so there’s some familiarity and maybe a bit of nostalgia there.
For some of the above, plastic works fine. Plastic is both soft and insulated, so there’s no problems there. However, it doesn’t have all that high of a heat tolerance, so it’ll melt if you’re not careful. Also, it’s not very strong, so it won’t do a good job scraping the bottom of a pan, nor will it hold up stirring something particularly thick. Finally, some plastics release chemicals when heated, and so if your plastic is not BPA-free, there may be some health risks from using plastic utensils in hot liquids.
Metal utensils work great on the strong and heat tolerance, but not so much on the soft and insulated.
Silicone is a good alternative to wood; by and large, it will handle 1-4 without a problem, if it’s a well-made utensil. Wood’s a bit harder than silicone, so you might have an easier time scraping with a wooden stirrer than a silicone one, but silicone is a whole lot easier to wash and care for, so if you’re not really used to wood, and you’re not particularly nostalgic, it’s a good alternative. I know that I’ve probably offended some people with that suggestion, but I stand by it.