As always, context is crucial. Sometimes they can be used interchangeably and sometimes they can't. Why don't you post some sentences using them and we'll see if you've grasped any differences?
in this sentence: "The government has called in the army to the affected area, and they are... food to the people who have managed to get away from the flood by moving to higher ground". The answer is "handing out", why not choose "distributing"
in this sentence: "The government has called in the army to the affected area, and they are... food to the people who have managed to get away from the flood by moving to higher ground". The answer is "handing out", why not choose "distributing"