I don't know much about Buddhism, so please excuse this ignorant question: In what way do Buddhist beliefs come close to, or match, scientific knowledge or methods?
Not really. Buddhism however does encourage observation and analysis, as means to understand the world. The Kalama Sutra is probably the most oft-cited example of this.
Of course with a religion/philosophy as nebulous as Buddhism there's no short answer. Some Buddhists reject Western science and only regard Buddha's discoveries as the real truth. Some Buddhists are scientifically-minded but regard Buddha's teachings and discoveries as "beyond science". Others believe science will eventually prove the Buddha right.