The difference is that in English, we are much more inclined to use the possessive pronoun when the hand is attached to someone. If we say "the hand is on the table" (assuming it is a part of the body and not a hand of cards), it implies that we don't know who it belongs to and suggests strongly that it is disembodied, like something out of The Addams Family..
Does »Ihr Hand liegt auf dem Tisch. / Sein Hand liegt auf dem Tisch.« sound more 'natural' than »Die Hand liegt auf dem Tisch.«? Because atleast in the context of an English speaker the latter sentence implies it's a disembodied hand; Would it be interpreted similarly by most native German speakers, or just sound a little bit odd?
Someone must have been visiting The Addams Family
auf, like a number of other prepositions, can take either the dative or the accusative case, depending on the meaning -- with the dative case, it indicates a location ("on"), while with the accusative case, it indicates the destination of motion ("onto").
Here, the hand is not moving; it is simply lying "on" the table.
I assume the former sentence was one about putting plates "onto" the table (they were moving from some other place and ended up on the table).