Tendon boots tend to be open at the front - they are usually used for jumping and allow the horse to feel if it knocks a pole, so hopefully the horse does not become blase about knocking poles because the front of its legs is protected. They protect the back of the leg - where the tendons lie, from being struck by the hind foot, which can be a nasty injury.
Brushing boots wrap right around the leg and mainly protect the inside of the leg from being knocked by the opposite foot, known as brushing.
Brushing boots go the entire way around the leg to give all round protection from brushes and knocks etc. They tend to be used to lungeing, hacking and schooling, and some people use them for XC.
Tendon boots have open fronts, but still protect the important parts of the leg. They also keep the leg cooler because of this. People tend to use these for jumping along with fetlock boots on the hind legs because it means the horse can feel it if it touches a pole, but still be protected from any bad scrapes and brushes etc.