I think that having money that obeys the rules of gravity is more realistic in the real world.
However, in a game world, I am perfectly fine with an abstract means of payment. And I agree, there are no actual banks and you're not actually making a wire transfer for your brand new sword and you are not paying that new breastplate with a cheque (US: check) or so, and it is not very realistic by itself, but what would that add?
Having money with weight requires rebalancing of the game in many ways, and it is not trivial. Low strength characters for example could get heavily penalized by this mechanic, as they could even get in trouble paying for this one really expensive Fabergé egg they for some reason desire, because it costs more money than they can actually carry. Furthermore, I would detest having to haul cash back and forth all the time, because it's, well, 100% boring.
Alternatively, the weighted money could be balanced by introducing a system of banks etc. But having a realistic system here (which is the main reason for having weighted money in the first place) would be too elaborate I think, and the benefits would not outweigh the meta-game that you now have to play, which distracts you from the actual game. I dislike managing my finances in real life as it is, please don't bother me with it in a fantasy game.
Of course, there are a lot of compromises between having an accurate and realistic economic system, and weightless money. But the thing is, in the end they are still compromises, meaning that the realism will have to be cut at some point anyway, and I am personally content when that cutoff is made at "weightless money".

Having weightless money results in having quite the same principles as you would when paying through banks or some other alternative means because you can't carry your cash. The difference is that you're not actually performing all the steps in between to get your money to the salesman across the counter. I consider that a good thing.
And hey, I wouldn't mind if your money had weight and it was stored in your vault at the end of time. It could automagically travel back and forth between the vault and your pockets. Money at the speed of light, so to speak, but that would mean it has to have zero rest mass ... d'oh.
