Participate = to join together with other human beings, for (whatever). To engage = can be used as a synonym, but it also means other things, two common ones are a) to engage in war / to engage the enemy. / to engage a hostile threat. b) to engage two different parts of a machine = make them come together so that there is movement. You will hear engaging of planes by military aircraft when they fly close to the presidents plane, or wander into a country airspace. You engage a clutch on a car or machine to move forward. https://en.oxforddictionaries.com/definition/engage

If followed by the word 'in', they mean the same. To engage in an activity is the same as to participate in an activity. However, the word 'engage' without the modifier 'in' can have several different meanings that are not the same as participate.