I would not anticipate significant differences with regard to the actual skills. Progressions vary between countries, but the fundamentals are the same. I don’t expect The adjustment to be difficult.
I image that the financial aspects will be the biggest difference. Everything is a business here, so expect to have to pay a lot of money for absolutely everything. Prices for training vary dramatically by region. One thing to know is that the monthly tuition is only a fraction of the overall cost. Gymnasts also have to pay a lot of extra fees: competition fees, coach fees, uniforms, choreography fees, and variety of other arbitrary fees. This can add up to an extra 2-5k annually.
Training practices are not standardized in the US. Each gym is independently run, and the owners get to do things however they want. A lot of coaches are immigrants and they bring diverse approaches to the sport. Where I live, there are American-style gym and there are Eastern European-style gyms, and they are quite different. The variation in gym cultures makes finding a gym that is a good fit for your child very important.
Good luck!