Are postcards and business cards really that important? Absolutely. But are they a requirement? No.
Having a set of postcards and business cards on hand wherever you go is so important. It's infinitely better to just pull out a card with your info than to try to write down your information on the back of some scrap paper when someone asks you for it.
The reason I usually recommend business cards over post cards is just for convenience-sake. But as I'm a guy, I'm usually not carrying around a bag that can fit postcards. I've found that women are more likely to have the postcards, and guys are more likely to have business cards which can fit easily into their pockets.
It's fine to have both but make sure to at least have one or the other because oftentimes the opportunity won't present itself for you to take out your large 8x10 headshot. It's also rude to give someone something that they will have to hold if you're at some sort of event. This is why business cards are the way to go in my mind... People won't get annoyed holding a headshot or postcard the whole time. Sometimes they might even toss it before going home because they're tired of carrying it around.