In P.E we are taught to stretch before we exercise as a warm-up to the muscles and let them get heated before hitting it hard at whatever it was you were going to do that day. The gym applies the same principles and when it comes to cool downs; same thing you stretch at the end. Believe it or not, stretching or basic western style stretching can do more harm than good and it's not always a good idea to get injured before you train. I believe in loosening up the joints more than the muscles because if the joints are better prepared, the muscles will have a better chance at making your workout that much more efficient.
I'm not an expert at anything but after nearly 20 years of experience in fitness I have the right to a opinion on what makes workouts more efficient or not. I want the workout to go as quickly as possible without losing form and staying at a consistent pace. Breathing is a major factor and how you breathe can make you or break you in your training. Yes I believe some good solid stretching is good but don't take so damn long you don't want to go full bore in your workout. I rarely if ever "warm up" anymore because I listen to my body as I train, I don't separate a little routine just so I can be prepared. Everyone works differently and some need to stretch before training if they're inflexible, in the beginning stages or need to build a foundation before doing anything remotely hardcore. A lot of exercises I perform have stretching automatically like Bridging, Animal Exercises, VRT and other methods of training. Now if you're a maniac and lift heavy fucking weight often and need to stay mobile yes you should stretch before and after training usually for recovery purposes not for the sake of being able to touch your toes.
Cool downs are usually a form of stretching or "extra" training to recover from the hardcore workout you just put yourself through and I have mixed feelings towards that. One way I look at it is that if you stretch too much after training and not understand the concept of deep breathing and naturally stretch you can pull muscles that can affect your next session and you won't be able to go the way you'd like to. On the other hand; if you can make a habit of breathing deeply and letting the body relax as you move or stretch after a workout this can be a great asset to your overall training because you're giving your body the recovery it needs and give it that little edge to make your next session even more bad ass than the next.
So are warming up and cooling down really that important? Yes and no depending on the type of training you do and the essential recovery you need to proceed to the next workout. Better yet, make your warm ups and cool downs part of the workout itself as you loosen up then hit it hard than relax and be loose, almost like moving meditation. I train every single day whether for 5 minutes or an hour and I have not had any major injuries or issues that are worth going to the hospital for and my flexibility has actually gotten better as I got older since I do hardcore type workouts and recovery workouts where I practice deep breathing and qi gong to stimulate blood flow and give my muscles and joints some relaxation type exercises which helps me stay in awesome shape no matter what I do. This works for me so whatever you wish to do make it work to your advantage and stay as injured-free as possible no matter what method you do; if you do things that are a set-up for injury it's best to avoid that method as much as possible. I believe in long-term training and being able to do things that save me from torn shoulders, ripped tendons, shattered bones (had enough of that in my lifetime) and doing what I love to do. Training is a lesson that you learn every single time you do something for a period of time; something to think about.