Obviously, one is a sphere and one isn't, but would the inhabitants of either world be able to rule out the possibility that their world might actually be the other?
We humans knew that our world was spherical because of its curvature, but the larger the sphere the less the curvature of its surface so it stands to reason that in an infinitely large sphere there should be no curvature (or rather it is infinitely small and indistinguishable from none).
Since they can't coexist with infinitely large worlds, assume that either there are no celestial bodies (sun, stars, other planets etc.) or that they depend on the location of the observer (there will always be a sun visible exactly above one's head, or perhaps the world has a luminous "ceiling" above the atmosphere)
Assume no space travel but arbitrarily precise measuring instruments and arbitrarily high resources dedicated towards finding out whether their world is a sphere or a plane.
Assume earth-like conditions (gravity etc.), on the sphere gravity will pull towards the center, on the plane it will pull down.