The differences are small but big at the same time, the body position varies slightly and how you use your body and weight varies, the biggest difference is the amount of contact you ride with in both hand and leg, though it does depend if you are doing dressage or HUS just like there are differences between a pleasure rider and barrel racer
Probably the reason the school starts everyone western is it is a little bit like the difference between sitting in an arm chair vs on a stool, one requires your balance to be a little more refined, that is not to say western is less refined at all (I've done a lot of both and know that is 100% not the case) just that the nature of the saddle is a little more forgiving to beginners
English saddles are a bit less so, I've seen many an experienced western rider hop on one side and keep going off the other as there just isn't so much to guide you in to place
I've also had a hunt rider get on my western trained horse and get lope, stop, lope, stop all in about 6 strides and she nearly came out the front door and the back and blamed the horse for misbehaving when actually her aids were just stronger than what he was used to and he reacted exactly as he had been trained to do, to body weight changes and light rein and leg aids (that would be normal 'contact' for a hunt horse)
Which western discipline have you been focusing on? and which English discipline are they offering you to try? I might be able to give some more specific difference examples for you