We use Solids often for Buffalo hunting, which is often undertaken in some real thick cover. These Solids will often go through unseen foliage better than a soft.
Naturally a good option for penetrating from the rear end of large game as it flees, often a required "follow up shot".
They are also often used on the small Game if damage in not preferred. In saying that, this lack of damage can also mean Large game will often travel further once shot as a small diameter hole will not allow blood and air to flow fast, often blocking up with congealed blood or bits of flesh, then you may indeed need to follow a wounded animal for some time..
The new Solids with a slight "Cup" on the nose are a good compromise on most game except Elephant. That is providing they feed in your rifle...