I can only speak to how these phrases are used where I live on the east coast of the United States.
WAIT A MINUTE usually means you want someone to wait briefly for you, but it can also mean that you disagree or are not sure about what the person is saying.
Examples: Wait a minute, I have to find my umbrella before we go out.
Wait a minute! Did Obama just say what I think he said?
WAIT ONE MINUTE is usually stronger and means that you object to what someone is saying or doing. BUT if you literally mean that someone should wait for you for about a minute, you can use this phrase also.
Examples: Wait one minute! You did not wash your hands after using the bathroom.
Wait one minute, I have to get my passport from my desk.