Let's say you want to walk instead of run. On keyboard, this is done either with a key that toggles between run and walk speed, or a key that is held in conjunction with the move keys which slows down your pace. So you get a wider range of movement speeds with controller. With mouse and keyboard, you won't find more than 3, and 2 is what's common.
For free look in a normal third person game, PC games will achieve this by holding either a key or a mouse button and then moving the mouse. This mouse movement will instead move the camera instead of your character.
You should try out ArmA 3 one day. It is a military simulator type game that showcases some of the most advanced and diverse movements, and has far too much functionality for a controller. It also lets you bind functions to macro key combos, like double clicking a key or pressing two keys at once.
If you look up videos of the game Dishonored, an advanced stealth/action game, you can do so much more in games like that with mouse and keyboard. That game on mouse and keyboard, especially at 120 FPS with blur reduction, showcases a level of fluidity that is mind boggling and unlike anything else. Shadow Warrior as well, but it doesn't have the same kind of advanced movement or platforming. Sleeping Dogs melee combat is another game in which mouse and keyboard unexpectedly comes in handy so much, being more intuitive than controller.