In modern French, the passé simple is almost never used in everyday speech. The passé simple is almost exclusively used in formal or literary written French, and most likely in the third person.
In other words, in spoken modern French, you will probably only ever need to use the passé composé to describe an event in the past.
To answer your question, however, here is the formal difference between the two past tenses:
The simple past describes an event generally in the remote past at a definite time, and that event has no consequence today.
E.g., La bataille de Marignan eut lieu en 1515.
On the other hand, the passé composé describes an event generally in the recent past at an indefinite time, which may have a consequence now.
E.g., J'ai trop mangé ce midi.
Il a emprunté une importante somme d'argent.