The difference between a wide angle fisheye lens and a wide angle rectilinear lens is equal area projection versus straight line projection. Uncorrected, they will both demonstrate field curvature.
All simple lenses will demonstrate field curvature based on the angle of view the lens provides. Of course the sensor/film size is also involved in the angle of view yielded by a particular focal length. When used with a 35mm sized full frame sensor, an uncorrected long telephoto lens, such as a 400mm one with a field of view (FoV) of only 5° will have field curvature the shape of a 5° arc of a sphere. An uncorrected lens with an FoV of 45° such as a 50mm one will, likewise, have field curvature the shape of a 45° arc of a sphere. As you can see, by the time a lens such as an 8-15mm fisheye is considered, the FoV on a FF sensor approaches 180° and the field curvature of an uncorrected lens would be the shape of half a sphere!
Most lenses used by modern photographic equipment are not simple lenses. They are compound lenses with several elements combined into several groups. Most of the additions beyond a simple lens are to allow for things such as close focusing, zooming, and to correct optical aberrations. One such aberration that is usually corrected to one degree or another is field curvature. This is fairly straightforward on a lens with a narrow field of view (FoV) because the curvature is much less pronounced than with a lens with a wide field of view. How much, if any, of the field curvature is corrected depends on each individual lens' design.
A lens such as the Rokinon 8mm T3.8 Cine Fisheye for Canon is corrected very well and yields almost a flat plane of focus when used on an APS-C sensor for a diagonal FoV of around 167°. At the other end of the spectrum, a single meniscus lens with the same FoV would have a very pronounced field curvature. Most designs are somewhere in between.