There appears to be two sets of card text on Howling Mine, depending on when it was printed. Do they function exactly the same? Or does the old one not matter if it is tapped or not.
Different printings of cards all work exactly the same way. The functionality of a card is determined by its Oracle text, which can be found on Gatherer (rule 108.1). For example, the Gatherer entry for Howling Mine says that the card text is
At the beginning of each player's draw step, if Howling Mine is untapped, that player draws an additional card.
You can see in your image that the old printing of Howling Mine has the type "Continuous Artifact". At the time, that meant that the card would only work while it was untapped. Since then, they have gotten rid of that type, and made the functional aspect explicit in the card text. This is why two printings have different card text.
In general, you can find many cards with text that has been updated over time to keep up with revisions in the rules. No matter what printing you have, the functionality is determined by the current Oracle text on Gatherer.