Began is generally not used with any kind of helping verb; it is a simple past conjugation. Begun is the past participle of begin. It always needs a helping verb to form a correct sentence.
– The movie began thirty minutes ago, and we are still not there yet./The movie had already begun thirty minutes ago, and we have not reached there yet. – It began to rain./The rain has already begun. – You began running as soon as you saw the dog./You had begun running as soon as you saw the dog.