Charlie's answer is excellent, but to append some technical information from my background as a civil engineer may be helpful.
When evaluating stormwater impacts for a site, there's generally 3 key criteria: runoff rate, water quality, and groundwater recharge.
Runoff rate is what it sounds like and is a function of the time of concentration, which refers to how long it takes a drop of water to travel from the most hydraulically distant point in a watershed (meaning what takes the most time) to some common low point (i.e. an inlet, a river, or some other receiving water body). The time of concentration is heavily impacted by the surface condition that the water passes over. Moving through thick grasses means that a drop of water has to keep changing it's path and thus has a longer time of concentration. Conversely, were water passing over an area of bare ground without any grass or other vegetation (such as after a fire), it will move much more quickly. The time difference can be drastic, some sites I've worked to develop have had the time of concentration reduced from 45 minutes to just 6 minutes; this means that water is hitting the receiving water body much faster than normal and thus causing much higher peak flows unless we design control basins to delay that flow.
The next criteria, water quality, is obviously impacted as you've demonstrated with your picture of a waterfall running black. Typically, water quality is improved by letting it infiltrate into the ground, slowing it down to permit sediment and fines to settle out, or some other method, but none of those techniques are available when water's moving fast with a high sediment load, hence, waterfalls that smell like extinguished campfires.
The last criteria, groundwater recharge, also suffers for most of the same reasons that the other two do: The water is just moving too fast to practically infiltrate into the ground. It's possible that the charcoal fines are impact the soil media to make it less permeable, especially if aspects of the fire 'compact' the fines into the media, such as vaporizing into the soil media and condensing therein as described in Charlie's answer.
In addition to all these issues, the reduced time of concentration means that the moving water has a lot of energy, which will increase the likelihood of ruts and gullies forming.