Alastair,
The other folks have posted have alluded to different land management agencies in the US and availability of wild camping. The vast majority of public land in the US is owned and managed by the Federal government. Three main agencies have jurisdiction over land that the general public may access without too much trouble (The other agency holding major land areas is the Defense Department, and their holdings are almost all closed to public access with special permits.)
The Interior Department manages National Park, Monuments, Conservation Areas, and BLM lands. The Department of Agriculture manages National Forests. National Forest are generally open access and you can camp in most places without problems. Some forests or portions of forests require usually easy to obtain permits for camping in certain areas. The same goes for BLM lands, which may have other activities going on, such as resource extraction, grazing and so forth. But vast areas of the National Forests and BLM lands are without restrictions. As another poster mentioned, National Parks are the most restrictive and most if not all require some form of permit for camping, although in many of the parks, once you are away from the road, you can camp pretty much wherever you wish as long as you obey rules about clearance areas around lakes, streams and other sensitive locations.
State Forests vary in their rules, but are often the largest pieces of public lands in the non-western states, although there are some largish National Forests in the east as well.
Your best bet is to get a land use map for the areas you wish to visit and by seeing who owns what, can make reasonably accurate assumptions about where you may legally camp.
And then there is always stealth camping…
By the way, Amy and I have backpacked extensively in Great Britain and have much appreciated of the live and let live attitudes about camping in England, Scotland, and Wales. We have never been hassled and have generally just discreetly put our tent down wherever we ended up for the day.
James for Amy