How large of a memory card do you need? A few years ago, a one gigabyte card was considered quite large enough; these days, 16 GB cards are popular and the recently introduced SDXC standard will allow cards to hold up to 2 terabytes!
Some of the trade-offs are obvious: smaller cards are cheaper! If you have a limited budget, buying a larger card may mean settling for slower transfer rates.
Ignoring the money issue, the advantages of a larger card are obvious: you don't have to swap cards as often or worry about running out of space during a long day of shooting. You also don't have as many things to carry; if you're using a half-dozen memory cards, it's much more likely that you'll get them mixed up or lose one than if you only have one or two. You also put more wear on the camera door changing cards more often.
Of course, the flip side of that is that when you have more cards, if one of them fails, you lose fewer photos. You may also be able to get the same amount of space for less money; two 32 GB cards will generally cost a lot less than one 64 GB card!