From B787 FCOM.
What is the difference between clearway and stopway?
The definitions linked in the question for a clearway and stopway are fairly adequate.
Clearway - is an area extending beyond the runway end available for completion of the takeoff operation of turbine-powered aircraft. A clearway increases the allowable aircraft operating takeoff weight without increasing runway length. No object or terrain may protrude through the clearway plane except for threshold lights no higher than 26 inches (66 cm) and located off the runway sides. The area over which the clearway lies need not be suitable for stopping aircraft in the event of an aborted takeoff. A clearway must be under the airport owner’s control, although not necessarily by direct ownership. The purpose of such control is to ensure that no fixed or movable object penetrates the clearway plane during a takeoff operation.
Stopway - is an area beyond the takeoff runway centered on the extended runway centerline and designated by the airport owner for use in decelerating an aircraft during an aborted takeoff. It must be at least as wide as the runway and able to support an aircraft during an aborted takeoff without causing structural damage to the aircraft.
Opa-Locka Executive Airport (KOPF) has both a clearway and a stopway at the end of runway 12:
The 1,000 ft of clearway extends to the end of the area. The stopway is shown with yellow chevrons.
Why is clearway minus stopway used in V1 adjustments?
To answer this question some background information must be given. A clearway is used for turbojets where the airplane manufacturer provides accelerate-go distance calculations. A stopway is used for turbojets where the airplane manufacturer provides accelerate-stop distance calculations. Neither clearway nor stopway distances can be used for turbojet aircraft that use a balanced field length concept.
Aircraft with unbalanced field length calculations can use the clearway for accelerate-go calculations and the stopway for accelerate-stop distances. The idea is to allow the aircraft to increase their maximum takeoff weight by utilizing those airport features.
A change in V1 speed has the effect of increasing one of the takeoff distances while simultaneously decreasing the other takeoff distance. For example, if V1 is decreased, accelerate-go distance is increased while accelerate-stop distance is decreased. There isn't an easy way to calculate a max allowable takeoff weight (MTOW) by adjusting both V1 and aircraft weight.
To make the calculation easier some practical things are implemented. A stopway will always be equal to or shorter than a clearway. This effectively increases the length of the runway to the end of the stopway for performance calculation reasons. At this point, we can go into the performance charts for both accelerate-go and accelerate-stop and come up with a MTOW. Since both calculations are using the same distance we need to adjust the accelerate-go distance by the difference between the clearway and stopway. We can then adjust V1 by the appropriate chart in the aircraft AFM by that difference.
We can perform an accelerate-go and accelerate-stop distance calculation again to come up with a corrected MTOW.
The reason the V1 adjustment works in this fashion is because V1 is the decision speed at which the aircraft can continue on the runway surface remaining, get airborne, and climb to the (adjusted) screen height with one engine inoperative before reaching the end of the Clearway, or stop on the remaining runway surface plus any Stopway. The function of "Clearway minus Stopway" is because the aircraft can be airborne and climbing over the Stopway in the "Go" case, but can't be on the ground, ie. it has to get airborne before the Stopway commences. If the RWY has lots of Stopway, but no additional Clearway beyond (the bottom rows where Clearway minus Stopway is a negative number), V1 must increase, because the aircraft has plenty of Stopway to stop, but must be going faster to "Go" and make the screen height by the end of Stopway.