To start something = shuru karna, usually a subject and object is required. To get something started = shuru hona = The subject is not needed, like passive voice. Examples: Mai ne kaam shuru kar diya he - Subject = main, Object = Work Kaam shuru ho gia he = The work has been started (may be automatically, the subject is not mentioned, who started the action).

both are most likely same

Shuru karna means something is not started yet,but just about to start. ex-khel(game) Shuru karo!!! (let the game begain !) . Shuru hona means something has started and now going on.. ex -Prathna(prayer) Shuru ho gya. (Prayer has started.)