Both have standards for road construction (lane width, road bed, shoulders, etc.) The Interstate Highway system is more strict than the US Highway system (2 lanes each way, divided highway, limited access, and better road beds, etc.) A US Highway may be anything from a 2-lane road with shoulders to a limited access, multiple lane highway – and may change from one form to another over it’s length – even within the same city.
US-66 is probably the most famous US highway.
Numbering of the two systems is “opposite” in order to reduce the number of times the same number is used twice in the same location. For instance, US highway numbers increase from north to south, and from east to west. Interstate highway numbers increase from south to north, and from west to east.
As other people have said, route numbers (both Interstate and US highway) may be overlain on the same highway. (For instance, I-75 and US-23 in the northern parts of Michigan, I-275, I-96, and I-696 in SE Lower Michigan, I-80, and I-90 across northern Ohio and Indiana.)
Standards change over time, and as the US Highway system is older, there are more routes that are on highways that would otherwise be considered substandard currently (narrow lanes, poor entrances and exits, etc.) There are even a few instances where there are (or have been) draw bridges (and requisite stop lights) on Interstate Highways.