Kidding aside, as I understand it (I'm certainly no expert), the terms most often used are biased versus unbiased rendering, were biased has artificial limitations, while unbiased employs 'real world' calculations.
So why use biased renderers ?
Well they can typically create a very good result using much less time compared to a unbiased renderer, on the other hand they typically also require that you mess around with a lot of knobs in order to get good results, meanwhile with a unbiased renderer you can just set up a 'real world' scene and it will render as such (albeit more slowly).
My guess is that Renderman is the most widely used biased renderer today, with Arnold being the most used unbiased.
You can have biased physically-based rendering and you can have unbiased 'not quite physically-based' rendering - in the latter case for example, it's possible to render direct lighting only (so no secondary bounces, or global illumination), which while obviously not the correct real result in physical terms, to the extent that you're evaluating direct lighting only the result is technically unbiased in terms of light transport. Similarly, it's possibly to have a spectral renderer (which in theory should be more accurate) which is biased, and a non-spectral renderer (RGB only) which is unbiased.
Biased can mean things like taking short-cuts or approximations - e.g. irradiance caches for diffuse results, or caching occlusion in order to slightly bias which light to sample per vertex in order to sample lights more efficiently: Both of these biased techniques generally give faster less noisy renders, but it's possible you might not notice they are biased in terms of the effect they have on the render - it all depends on the scene, materials and the lighting. For simple scenes you probably won't notice, for more complex scenes with nested medium materials with glossy/pure specular responses or with refractive caustics, it's very likely you will notice the effects biased rendering has compared to unbiased rendering.
Renderman RIS can be set up to be unbiased (but with default settings - radiance clamps) it's not. Similarly Arnold's default settings of having a light threshold (under which it won't sample a light) is also biased, but obviously this setting can be changed. Arnold can also cache diffuse contributions on hair, which is also biased.