The difference can be seen in two board games: Pandemic and Defenders of the Realm. Pandemic is a game that uses cards to determine where one of four diseases will erupt on a map representing the whole world; Defenders is a game that uses cards and dice to represent where one of four armies will appear on a board representing a fantasy realm. If they sound similar, that's because they are: Defenders was specifically designed to mimic Pandemic, except with a fantasy flavor, and with dice.
Of the two, I prefer Pandemic, because the cards are a resource that must be managed, and there's no hoping for a lucky roll. Even though the card draws are random, the astute player will notice when the action pile is running low, and when particular city cards are more likely to turn up. Thus the game has more of a sharp, strategic feel to it.
Most RPGs use dice, and it's easy enough to see why. If you roll a successful hit, say a 19 on a d20, the pool of future successful hits isn't diminished. (Primetime Adventures uses cards, but then shuffles them back after every scene; it doesn't take advantage of the traits that make card games what they are, and could just as well be represented by a dice pool mechanic.)
As an evening's diversion, I once had my players roll out five combat rolls at a time and write them down on index cards, e.g.:
4 16 3 12 10
...they'd then tick off a roll that they wanted, knowing that it meant not being able to get that score later:
Although "gamey", and we didn't do it again, it was a fun and interesting way of injecting a little strategy into what would otherwise be a blind die roll.
It made me think of what would happen if I set up a fortune teller's booth in the town; if the players visit, I'll make up some dramatic hokum with cryptic predictions and mumbo-jumbo about tea leaves or tarot cards, but then hand them a strip of paper like this:
6 16 20 4 15 19 11 5 9 11 3 5 16 9 8 17 13 2 12 19 12 9 10 20 14 1 5 2 12 8
...only this time, they don't get to choose the order. They may hold off going up against a frost giant until they can burn that low roll off on a kobold, however.
Alternately, you could have 6, 8, 10, or 12 of these made up, and roll an appropriate die to select them. And there's no need to restrict it to combat rolls; they could just as well be saving throws, although something tells me it wouldn't do to mix them...each strip would be for one roll type.
Just a couple of considerations for altering the flavor of the randomness in your games.