I've ridden both, but I personally prefer to ride English. I feel much more like I'm doing something productive rather than just sitting there going around in circles. (Now may be a good time to throw in that I jump?) I tried jumping in a Western saddle once, never again. I got stabbed in the stomach with the saddle horn and then it went forward kind of under my rib cage and tugged down..fun stuff. Nobody was hurt though, so no big deal.
Anyway, I prefer posting to sitting. It makes me feel more comfortable and like the horse is more comfortable. I love the smaller saddle, it lets you feel closer to your horse and you don't have anything to stab you if you jump, your horse rears, or he bucks and your fall forward. You have shorter stirrups which, in my opinion, allow for better control. I have ridiculously long legs so my heel is usually under or level with the horses barrel in a normal saddle, I'd look like a giant with my leg straight (Western OR dressage). I cannot stand riding with one hand..at all. The other one just lays there limp doing nothing productive.. I like direct reining (contact with the horse's mouth) much better than neck reining (horse knows when to turn when reins hit his neck a certain way). Don't get me wrong, on a good horse I can drop them altogether and ride off my legs I just really dislike neck reining..
English really isn't very 'proper', we just have tighter pants and different show clothes. Really, if you look at a WP class, people have their hair and nails done, make up on, eye-burning glitter all over their clothes, etc. Sure, they look more relaxed but in my opinion the horses' strides look painful. I like big horses who step out.
Just because your moms friends horses are trained Western and you're being trained English doesn't mean you can't ride them, it's just a matter of asking their owner how they want the horse ridden and such.
So differences:
English:
Smaller saddle
Shorter stirrups
Direct reining
Should be forward moving
You can jump
Breeches, a polo, tall boots, and a helmet as show attire.
Usually post to the trot
Western:
Bigger saddle with saddle horn.
Longer stirrups
Usually neck reining
Can be forward moving (Barrel or reining horse) or can be slow and dead-as-doornails (WP horse)
You shouldn't jump, or at least not very high.
Jeans, chaps, a fancy long sleeved shirt, boots, and a cowboy hat as show attire.
Usually sit the trot, though you may post. I don't think Western riders ever do 2-point.
Be glad your mum is taking you to lessons! Lol