The main one: Barns vs. Agistment.
From what I've read on here, if you don't have your own property in the US, your only option for horse ownership is to keep your horse at a privately owned barn, wether it be pasture board, stall board, or a mix of both. There is a BO that governs the way things are run in the barn (and fair enough) and often disagreements arise due to rules, other boarders, etc. Barns can have their own show teams, and are often discipline specific, I.e. Hunter barn, Western Pleasure barn, etc. Barns sometimes will have a resident trainer for a trainer who regularly gives lessons at the barn. Barns also tend to have their own facilities, I.e. Arena, (indoor/outdoor) round yard, wash bay, tack shed etc.
Compare that to the most common method of horse ownership in Australia - Agistment.
In cities (at least my city) there are government owned agistment centres where people can 'rent' paddock space. You get a spot amongst 10-30/40 other horses in large paddocks, a couple of yards for sick/foundred horses (amount/quality varies) and in some places rudimentary facilities - enclosed paddock for riding, some places have outdoor arenas, basic jumping equipment, etc. No place to keep your own gear or feed. Managers of these paddocks are only there for uokeep of pasture and fences, they never interfere with the actual horses, that is the owners responsibility.
There are also privately owned agistment complexes, which are more similar to the American 'Barn'. There are normally a few stable blocks, many paddocks, indoor/outdoor arenas, round yards, holding yards, wash bays etc. various types of agistment on these places, private paddock/share paddock, Stable & paddock combinations, Partial board (they feed/rug in the morning, you do at night, etc) and ful board (all feeding/rugging/stabling done for you) These places often have riding schools based in them, and/or resident trainers and many host their own events.
I don't think I have EVER heard of a horse being on pure stall board in my city, I think it is a more common occurance in the US. The majority of horse owners I know keep their horses in Government paddocks as explained above for around $20pw.
Many people from the US are flabbergasted that my horses live in a pasture 24/7, and don't have a shed/stable to go into. Over here it's the norm! I think my horses would go nuts locked in a stable for any more than night time, I like to think they enjoy their 'horse' time.
Another difference: Rugging.
We have a huge ruggng culture here in Aus, and I don't think it is very big in US except for show horses.about 80% of horses I know in my city are rugged in one form or another through winter, and probably 60% year round. For us, rugging regulates coat, warmth, wind/rain protection in winter and sun/insect protection in summer. Many horses here are clipped in winter and rugged to allow for heavy workloads.
From what I know unless you have show horses ruggin is much rarer in the US. When I was over there last year in April/May, the horse I rode was clipped and not rugged, which is nearly blasphemy over here!
And another, BIG one: Trainers.
Common questions here: Do you have a trainer? Ask Your trainer. What does your trainer say? Take him to a trainer.
I don't personally know anyone in my city who consistantly works with a trainer. Part of it is accessability. As most people don't keep their horses in 'Barns' Trainers are hard to find and expensive to use. The most training many kids get is from volunteer instructors at pony clubs and other youth organisations. Many others just learn by trial and error. Another factor is that trainers in certain disciplines can be hard to find. There are plenty of Dressage/Eventing/SJ trainers, but find a campdrafting/cutting.cow work trainer in Canberra? Get real. The closest trainer to me who teaches my discipline is about an hour and a half away.
We still turn out as decent riders. We just learn from trial and error.
And a pet peeve of mine: The whole western Vs. English mentality in the US.
It seems that in the US, trail riding is majorly a 'western' discpline. Cow work is Western. Rollbacks, good stops, haunch turns = Western. In Australia, you don't ask 'western or english?'. You ask 'what discipline do you do?'. Riding is riding. I think I know 1 family in the part of the horse community I am in with a western saddle. Yet we do cattle work, station work, campdraft, slide stop, spin, rollback. We also do dressage, SJ, eventing, on those same horses. We just ride. It is all horsemanship, I don't understand why it is such a big divide over there. Why should someone be looked down upon is they barrel race and don't have a western saddle? Over here the msot popular saddle for barrel racing outside of rodeo is a stock saddle, and guess what, the times are just as good :] So what if you want to practice dressage but don't use a noseband? It doesn't mean your horse can't do dressage or can't barrel race or can't chase cows because, shock horro, you don't have the right saddle. If you had to have the exact gear to ride horses, many of us should never had started riding in the first place.
Another thing that ties into this, the idea that your horse must be in a curb bit to neck rein. This idea is absolutely ridiculous! Sure, a curb will refine rein signals. But the whole point of neck reins is that the horse responds to the pressure of the rein on his neck, along with leg/body cues. The bit doesn't even come into proper neck reining!
Okay, I think that's about all i've got for now. Feel free to correct any misconceptions, error, whatever. And feel free to debate or discuss anything i've said :]