We says "wait a minute" when we want someone to slow down or stop, but we say "wait one minute..." when we are strongly disagreeing with someone and we want them to stop talking and listen.

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.

I slightly disagree with the other answer "wait a minute" can either mean you want someone to stop moving so you can catch up, or it can mean you want someone to stop talking as you disagree with them. "Wait one minute" is usually before a short journey (to the shop maybe) begins and you are asking your partner not to set off, but indicating only for a short time. But it's not a definitive difference and the tone used offers the meaning when it is said.