It actually depends on the usage. Although practically they are basically the same. However, "wait a minute" is sometimes used when you suddenly remember/realize something. For example: "Today is Sunday and I am going to church. Wait a minute, today is Friday not Sunday." Or if you want to stop/hold a person when talking, because you want to add or ask something. For example: (situation) A person is scolding you for "accidentally" spilling cola on his shirt. And you would interrupt him with: "Wait a minute, it's not like I did it in purpose." On contrary, "Wait for a minute", suggests that you really have to wait for 1 minute.

There is no difference.

"Wait a minute" is frequently used idiomatically in British English to indicate a slight pause, a hesitation or a moment of realisation, such as: "Let's go to the park - oh, wait a minute, I saw a nasty dog there earlier today, let's go somewhere else instead." These variants all mean roughly the same: 'hold on a moment/minute/second', or 'wait a moment/minute/second'. On the other hand, depending on context, it can have the same meaning as 'wait for a minute', which is a more direct statement -- wait for one minute.

There is no difference, but how you say either can make a difference in the meaning.