You'll do both things more efficiently if you give them your all rather than having your attention constantly flit between them. Imagine if, while programming, you had to switch between projects every five minutes. You'd never get anything done! Yet we expect our brains to perform well under the same harsh circumstances.
Of course, if you don't care about what you're doing, it doesn't really hurt to let your mind wander. You'll probably do a sloppy job, but you don't care anyway. But the real payoff to concentrating on something like washing dishes is that your mind gets better at concentrating, so you'll instinctively do it when it really matters. It's like muscle training — you don't want to wait until you need to heft a couch to start doing it.
