Your two sentences do not mean the same thing, though they refer to the same event.
The simple past designates a past event or state.
The present perfect designates a present state arising out of some past event or state.
This is why the present perfect is not used with temporal qualifiers which do not include the present: "They have begun cooking last night" is not acceptable English.
What you can say is:
George began cooking last night, and John began cooking ten minutes ago, but both George and John have (now) begun cooking. Jane has not yet begun cooking.
Perhaps the difference will be a little clearer if we use a telic verb rather than the aspectually complex begin:
George cooked dinner last night, and John just cooked breakfast. Both George and John have now cooked a meal. Jane has not yet cooked a meal: she will cook lunch today.