Rainforest and Jungle
There is a difference between rainforest and jungle, though this does not have much to do with rainfall. Jungle areas typically surround rain forest areas, and have a key difference that has little to do with climate. The biggest distinction is that rainforests have an incredibly dense canopy of very tall trees. This means there are far fewer plants at ground level. Trees block out much of the light that would be required by most plants to thrive, so you will usually only find shade loving ground plants in rainforest areas.
In contrast, a jungle allows some light in, and jungles are often located surrounding rainforests, near water sources like rivers. Parts of the Amazon rainforest are really jungles, because the tree canopy does not fully block the light. This can make the ground much more difficult to navigate since you will have more ground dwelling plants and shrubs that grow very tall and block paths. If you think of movies where characters are shown clearing their way through paths with a machete, they are typically in jungles and not rainforests.
As for lions being kings of the jungle or the rainforest, this is really inaccurate. Most African lions are kings of the savanna: large, open, grassy plains and spend little time in tree dense areas. Some nominate the jaguar as king of the Amazonian rainforest, since it is one of the top-level predators that dwell there.
Why do we find fewer ground plants in rainforests than in jungles?
Jungles receive less rainfall than rainforests.
Unlike jungles, rainforests are usually located near water sources.
In rainforest areas, much of the sunlight is blocked out by tall trees.
Certain animals such as lions and jaguars spend most of time in rainforests.