Cash is a decentralized system. Any given bill holds its value more or less independent from its surroundings, assuming of course that the government is still around. Credit and debit cards, on the other hand, form a centralized system - your credit card may have a fancy picture on it, but unless whoever you're trying to pay has a dedicated communication channel with the credit card company, it's just a very pretty piece of plastic. This distinction is made clear in several ways.
First, authentication of value: Cash uses a relatively weak distributed authentication scheme - the bill or coin only has to look valid, which can be made difficult, but never impossible. This only works because breaking the authentication (counterfeiting) on a large scale can be expensive, and each individual break only nets you a relatively small payoff (the value of the coin or bill). Credit and debit cards, on the other hand, use relatively strong authentication for value - you contact a centralized server, which can keep track of all currency in the system. This is far more difficult than the decentralized cash model, but it's also next to impossible to break, assuming the server is written securely. (Note that authentication of value isn't the same as authenticating the owner of the money - both cash and credit cards are pretty bad at this, in different ways.)
There's also the issue of robustness. It's not unheard of (even if it's not exactly common) for credit card processing to "break" at a given venue, forcing them to process all transactions with cash, or by other means. This can happen because credit cards simply don't work without being able to contact a central location to get the balance on the card, and subtract from it. Cash, on the other hand, is available for use under a much wider range of circumstances. Because it holds its value in a fully decentralized manner, it's not subject to any kind of communication breakdown, and so it works as long as people are willing to accept that it has value. (Note, though, that the authentication is weaker because of this - thus, counterfeiting is possible for cash, while a fake credit card wouldn't get you very far.)
Cash also has the property that, since no centralized communication is required for its use, it can be used without being tracked. In other words, cash can be completely anonymous. With credit cards, there's a single point through which all transaction data flows, and this in turn means that data about when the card is used can be collected very efficiently. This is another property that applies in general to centralized systems, but it can be applied to decentralized systems too - it's just harder. We could imagine, for instance, a system in which people were required to scan all currency that they handled into some sort of centrally controlled device, to prevent counterfeiting. This could be circumvented, since the scanning step isn't required for the transaction, but it's a way to graft centralized notions of control onto an otherwise distributed system.
This is an important point, actually. In a lot of cases, centralized systems can have decentralized aspects added, or vice versa. With money, for instance, it's all printed in a few centralized locations, and this contributes to it having value: if anybody could print their own money, it'd be worthless. When it comes to distributed systems, introducing centralization can be a powerful technique, if the tradeoff is worth it.
In other news, I haven't figured out everything I want to do for the rest of these, so I just might take requests. XD