What is the difference between the Maximum Glide and Minimum Sink ratios?
Glide ratio is the ratio of the distance a glider can travel horizontally to the altitude lost in transit. For instance, if a glider can travel 40 miles horizontally while losing one mile of altitude, the glide ratio is 40:1 (typical for a medium-high performance glider). The best glide ratio or maximum glide ratio is simply the best ratio a glider can achieve. Best glide ratio is achieved at a specific airspeed, which varies depending on the glider type. One flies at best glide speed in order to maximize the distance covered. When gliders are flying in a straight line and want to go as far as possible, they fly at best glide speed (typically 55-65kts).

Minimum sink is the minimum vertical speed the glider is capable of flying at in still air (typically 100-200ft/min). Minimum sink is also achieved at a specific airspeed, depending on the glider type (typically 40-50kts). One flies at minimum sink speed to maximize altitude gain when flying in lift. If you're circling in a thermal that is going up at 500ft/min, and your minimum sink speed is 100ft/min, then the glider will go up at 400ft/min.

To review, the key differences are:


Glide ratio specifies how "flat" the glide angle is
Minimum sink specifies the minimum vertical speed the glider can fly at
One flies at best glide speed to maximize the distance covered vs altitude lost
One flies at minimum sink speed to maximize altitude gain in lift
Best glide ratio is achieved at a higher airspeed than minimum sink


Any aircraft can be a glider with its engines out, so this applies to all aircraft.


  However, I don't understand how the lowest sink rate you can get isn't the same as the shallowest angle you can glide at.


Remember that minimum sink comes at a lower airspeed than best glide. At minimum sink speed, the glide ratio is worse because even though the aircraft is sinking slower, it is also moving forward slower.