They are two very different words with different meanings. Embark means to start a long journey. "I embarked on my mission to visit every country in Europe." Engage means to become involved with something. Its meaning varies though depending on how it's used: "Engage the enemy" is a common phrase meaning attack the enemy. "To be engaged to a person" means you are set to be married. "Engage in conversation" means to talk to someone.

embark: to begin (something that will take a long time or happen for a long time) She's embarking on a new career. The company has embarked upon a risky new project. http://learnersdictionary.com/definition/embark engage in [phrasal verb]: engage in (something); to do (something) At college she engaged in various sports. We don't engage in that sort of behavior. http://learnersdictionary.com/definition/engage As you can see, "engage in" is a phrasal verb. When we form a phrasal verb by adding one or more prepositions to a verb, it often changes the meaning of the verb. In this case, both "embark" and "engage in" mean to do something, but embark refers to activities of longer duration and the emphasis is on the action of starting/beginning the activity.