Someone asked me whether I was at home or not last week. When I say as a response, "I was away last week", or "I was out of town last week". is there any difference in the meanings? I just want to mean that I wasn't home, but didn't want to say the details. I'd like to know whether I can use these two sentences interchangeably or not. Also if there is any other way of saying, please explain to me.
Away just means away, i.e., away from home, but doesn't necessarily mean out of town. You might have been staying in town at a friend's house. But it sounds like you really were out of town, and thus you can use either one. They are both very commonly used phrases.
Of course, if you really don't want to give any details you can simply answer:
No, last week I wasn't at home.