Oh this is one I can answer, as I’ve camped here and at the places you’re talking about in Spain.
First, some terminology: we call this car camping (you may also see things about backpacking which refers to “camping sauvage”). If it’s a primitive campsite, there is probably no electricity or toilets (there may be a latrine which is a toilet over a hole, no flushing; if there’s no latrine, you dig a hole in a responsible place away from water and cover it up).
Our campsites are way more private than the ones in Europe. Rather than being lined up in a row, campsites here are usually arranged in a loop/circle and you won’t be right next to your neighbor. Normally they put the RV’s on one loop and the tents on another; there may be cabins too. Tent camping here is way cheaper than in Europe! Typical price in my area (Appalachian mountains) is $20 for a spot that can have up to 4 people; primitive sites are even cheaper. The common areas in general are way less clean than in Europe, especially the bathrooms and showers. We don’t have the nice washing areas that you have; you bring your own plastic tub and fill it at the water pump to wash dishes. (You should use soap designed for camping that biodegrades better, and make sure the soapy water doesn’t get into the ground and water supply.) We don’t have a bar/cafe onsite; some places have a general store for groceries and supplies, but it will be cheaper if you stock up at a grocery store - however, you should buy firewood locally to make sure you don’t spread tree diseases by accident. We don’t have pools at our campsites but you may be able to swim in a lake or river if there’s one nearby.
Most non-primitive campsites will have a picnic table with benches, a fire pit with a grill over it, a hook to hang things (trash bag, lantern, wet things, etc). You can have an open fire in the pit for cooking, or bring your own gas stove. Our outdoor areas do not have lights like in Europe, so I recommend a headlamp for nighttime. There will be a common area with the bathrooms and wildlife-proof trash bins. It’s very important in American campsites to throw away all your trash that has touched food, because we have more/bigger animals that will eat it; never keep food in your tent. If you’re in an area with bears, it’s not even safe to keep food in a car and the campground will have very strict rules and recommendations around making sure the bears don’t get any food from campsites.
Most people camping will also be going hiking, fishing, playing outdoor games, biking, or just hanging out in nature. Americans who are camping are generally very friendly and love to show strangers their setup, some small item they’ve brought that’s perfect for a task, or a recipe they’ve perfected. If you’re lucky, there won’t be screaming children running around. You may see a large group of school-age boys with a few adults; those are Boy Scouts, they generally are well-behaved and are required to have very good camping habits. If you need help or advice, the boys or their troop leaders will be very knowledgeable and eager to help.
So in general, campsites in Europe are more clean, campsites in the US are more private and put the focus on being in nature. Both are fun and it’s nice to go and see what other people’s camping setups look like. Both are good for people watching as well.