At the hotel checkin counter they will keep the transit cards. By the rules you're not supposed to leave the hotel or airport grounds. I guess the hotel gate will stop you from going to town. But if you're bound and determined to explore you can probably find a way, e.g. tell them you're going to the terminal to work on your reservation. And from there get a taxi to town. But of course if you get caught you could be in trouble.
I was picking my friend up from the hotel yesterday (she had a full visit visa) and they checked ID's at the gate when we entered by car and scrutinized her passport and visa very carefully before opening the gate/barrier. Telling them you are going to work on your reservation in the airport will not work.
Also in Jordan,they often stop cars (especially on the airport highway) and check ID's,is it worth it to get in trouble because of 40JDs?
To worldiswide,could you provide some more details so we can help you?