I own both (1 leather western, 1 synthetic western & a synthetic Aussie) and have owned a synthetic english before as well. My favorite is the leather one, but that is only because it is more solid & looks better lol. They are of course more expensive.
I have nothing against synthetic, however. I personally find them comfortable and most have flexible trees whereas alot of leather do not. This is a good thing because i've found they can fit a wider range of horses, but if you can find a good leather saddle that fits your horse & you well, they are nicer.
If you plan on competing, of course, then I would def. Go leather.
For trail riding and hunter paces, synthetic all the way. Very sticky saddles which buys you many extra lives when your horse does something unexpected. It's very hard to come off in a synthetic saddle. Also don't have to worry about getting caught in the rain.
Anything that involves better communication with the horse like dressage, or jumping where you have to be able to adjust your horse's stride, leather. Leather absorbs some of your motion and tends you give you better communication with your horse while synthetics tend to have more bounce to them.