What is the closest US equivalent to UK Ordnance Survey Maps?

Since you mention Yellowstone and Grand Teton National Parks, there is a National Park Service, a bureau of the U.S. Department of the Interior that provides simple and detailed maps of all of the national parks.

There are simple maps showing roads, buildings, park attractions etc. as well as topographical maps that include trails, landmarks, and most everything else.

• You can find all park maps here: http://www.nps.gov/hfc/cfm/carto-atoz.cfm

• Yellowstone maps here: http://www.nps.gov/hfc/cfm/carto-detail.cfm?Alpha=YELL#

• Here is a detailed map for yellowstone: http://www.nps.gov/hfc/carto/PDF/YELLmap1.pdf

State park information can be found on the individual state's website or the state parks website. For example from Texas Parks and Wildlife Department is TPWD: State Park Maps and Big Bend Ranch State Park Map.

Any other land area is likely privately owned and thus you may not be able to enter and there won't be a map showing paths etc.

The USGS maps are indeed not very useful for outdoor people. They are better than nothing, but lacking a lot of detail.

National Geographic Maps are available for popular recreation areas such as national parks. They are much more detailed than the USGS maps. Sometimes also local organisations/companies produce their own maps (e.g. this one for the Wasatch in Utah.)

The most significant differences between the UK Ordnance Survey maps and any comparable US maps come from several basic issues.

The UK (England and Wales, but not Scotland so much) has a huge number (more than 140,000 miles) of footpaths that have a historically and legally protected right of pedestrian walking access for the public.
Other paths are also protected (such as bridleways for horseriding, canal towpaths, etc), and there is also public right of access to some other areas as well as National Park areas. Although there are nationally recognized trails (such as the Appalachian Trail and North Country Trail) there is nothing in the USA that is comparable in mileage or complexity.

The UK has the Ordnance Survey that for more than 150 years has been responsible for mapping the entire country at scales of up to 25 inches to 1 mile. Maps - paper and electronic - are now readily available at scales of up to 2½ inches to 1 mile. They have a level of detail that includes not only footpaths and such but also things like public toilets, power lines, springs, archaeological sites and contour lines up to 5 meter intervals. Some products are freely available.

There is no equivalent organization in the USA.

Finally, whether or not you might be able to walk a path without compass or map, it is certainly true that without a map and arguably a compass you might be completely unaware of what is around you that you'd wish to avoid or go to take a closer look. If walking is nothing more than a trip from A to C via B, though, feel free to dispense with them.

Benchmark Maps are highly-detailed, colorful recreation maps of western US states. Periodically updated. Back of map contains printed info. Available through Amazon.