It really depends on which supplement you are taking. If it is a nitric oxide supplement with green tea extract in it for instance, at least from personal and anecdotal evidence you tend to get less exhausted during your workout and as a result you can do more volume and/or greater intensity. That comes from the way the body breaks down Nitric Oxide. The caffeine keeps you alert and raises your heart rate. Note though that taking it for long periods of time does tend to reduce your level of intensity during your workouts (Again anecdotal, not true for everyone). Taking it after a workout may help your body circulate nutrients better but don't quote me on that, I'm not a doctor. I do however know that some pre-workouts come loaded with L-arginine and some other amino acids which help with recovery time. So if you want you could take a pre-workout afterward and you can see some benefit but the question is why would you do that when you can get a supplement designed specifically for that purpose and save your money?
In reference to your green tea question, theoretically consuming green tea or green tea supplement has numerous health benefits (Vitamin C, helps skin, high fluoride content for you bones, helps lower cholesterol, the list goes on for days) regardless of your timing, so its not unsafe to say it may assist in your recovery. Consuming it prior to your workout will yield the results I stated above, essentially a better quality workout because of the caffeine content but you don't have the added effects of the nitric oxide. Consuming it afterward would keep your metabolism up after your workout is ended but its negligible if you are doing any kind of cardiovascular training as studies show your metabolism stays in a higher state for hours after your workout anyway. So at that point you would be taking it for the nutritional value and to stay awake. So no matter what you are benefiting. Know that it does contain caffeine and like any drug/herb your body gets used to it and will require more for the same effect; in your case the pre-workout alertness feeling.
As far as a difference between drinking a pre-workout versus regular green-tea, that is more of a personal preference. Taking a pre-workout consistently for a long period of time is not exactly great for your liver and kidneys depending on which brand you choose. You may also become dependent on it to get you through your daily gym routine which is also not good. Finally the affects tend to become less noticeable after extended periods of time. It is also more expensive than green-tea and does not have the same nutritional value. At least with green tea when your body adapts to consuming that level of caffeine you still benefit from the other nutrients in it. (That is not to say there are no nutrients in pre-workout supplements) Should you avoid a pre-workout completely? That is up to you but there is nothing wrong with keeping some around the house for days when you don't feel like training or you feel sluggish. Should you drink green-tea before and/or after? It is once again personal preference and there are numerous health benefits so it can't hurt your recovery.
So in summary, get both if you want. Drink your natural green tea every day (before and after is fine as long as you don't have heart problems) and keep some pre-workout around for those days when you feel lazy. Try not to take the pre-workout every day though.
I hope that helps, sorry it was so long but you asked a very multifaceted question.