Where to use involve or where to use engage: For example: They can involve/engage themselves more in the activities
When I try to explain it to myself, I recall the formula: “The difference between involvement and commitment is like ham and eggs. The chicken was involved, but the pig was committed.” 
Trying to be serious this time:
involve-- to have or include (someone or something) as a part of something. When you get involved in the Spanish Club, you attend meetings and events. The job would involve travel.
Involve is defined as to include or add as a participant. An example of to involve is a married couple including a friend in their drama (he’s involved).
engage – 1) to occupy the attention or efforts of a person\persons, to get someone interested: He engaged her in conversation.
2) hire: to engage a worker
I would advise you to turn to the BNC to get how they are used in different contexts.