Two simple oxfordian words that hold the key to what we do and how we do! I think these two words are absolutely not inter-changeable. Though in everyday usage they might convey the same message, with the latter conveying a more aggressive and on your face message. The former is the one I think more powerful.  Knowing as compared to believing in something is more powerful and more complete. Indeed it so sure of its existence that it defies any need of physical or tangible proof. Believing on the other hand can be subject of scrutiny, not by a second or a third opinion, instead by the believer himself.  
To give a simple example, we all know that we breathe, we don’t believe in it. It’s an end, without explanation, argument or substantiation. For the sake of argument someone might term this absolute stupid, because there are 7 billion nostrils sucking up oxygen from the air. But my contention is exactly the same, even if there were just 1 person breathing, he’ll still know it.   

The challenge that comes forth before us is to experience the same feeling of ‘knowing’ whenever we take any task. Such purity and absolute surety that even the idea of doubting is crushed beneath it.