Can we say a half hour in place of half an hour?

If no , then what is difference between two?
Usually, yes:

The show lasted a half hour / half an hour.

As an attributive adjective, however, only the first will do:

It is a half-hour show.

Use perhaps has to do with stress and ease of articulation in context:

I'll meet you in half an hour.
A half hour is not long enough for what I have in mind.