The Flat Earth is laid out like a North-Azimuthal projection.
The North Pole is at the center while Antarctica is at the rim. The continents are spread out around the North Pole.
Circumnavigation on an FE is achieved because on a compass East and West are always at right angles to North. Traveling Eastwards continuously takes you in a circle around the North Pole. East and West are curved.
As it happens, on an RE you do not travel perfectly straight when traveling East or West either. Consider this thought experiment:
You are standing 10 feet away from the North Pole. You are then directed to travel East and are instructed to continue to do so. What happens to your path? You end up traveling in a circle, and not in a straight line that you previously thought you would.
The exact same thing happens regardless of where are you on the Earth. Your path will not be straight without you having to constantly change the direction you are traveling in reference to a compass.
Traveling in a Straight Line
Q. Can't we just circumnavigate the earth by traveling in a straight line without a navigational aid?
A. It is not possible to travel in a perfectly straight line for very long without a navigational aid.
It's not even possible to drive down a long length of highway without turning the steering wheel left or right. Get in a car and see if you can drive down a long stretch of highway without turning the wheel left or right. It's a near impossible thing to do. And when it comes to planes, ships, helicopters, et all., no craft has the ability of traveling in a perfectly straight line without the operator adjusting the craft with regards to visual terrain, compass readings, or what have you.
When one navigates, hands on control is paramount. You wouldn't find a ship captain in New York pointing his vessel in the direction of London, turning on cruise control, and then taking a three week nap in the lower decks. Who knows where he'd end up.
Q. What about other types of navigational instruments?
A. Using a compass, gyrocompass, or looking at Polaris as a reference for Eastwards or Westwards travel will take the navigator in a broad circle around the North Pole.