If a person is dying of cancer, prays for a miracle, and watches the cancer go into remission, is that not evidence, to that person, of a higher power? (I am not a religous person, but bear with me.)

The scientist would say no, because the results are not repeatable. Science demands that they be so, and rightly so because science's goals are theory and fact. But religious belief does not have the same condition attached. Does this make it inferior? Only from the point of view of the scientist and his goals, because such belief is incompatible with those goals except as a means to inspire more effort toward them.

You contend that the two types of belief are different because of the reasons behind them, and I'm not saying I think you're wrong. But fundamentally different? I'm not so sure. They are both belief, and they both have an affect on the human spirit (by which I mean human dreams, aspirations, etc., not a literal ghostly spirit thing).

You say that one is rational and one is irrational, which is technically correct. I don't know if you meant any negative connotation to that word, irrational, though it often comes with that negative baggage. Illogical might be the word I'd use instead, for logic and illogic is a more cold, clearly defined division. Irrational suggests, well, crazy. I don't think there's anything crazy about belief in something despite a lack of evidence. That sort of belief can have very real effects on the human condition, on hopes, dreams, interactions with others, the things that inspire us, the things that make us feel wonder and mystery. Those aren't irrational, even if the belief technically is. Sometimes that illogical belief can lead to negative effects, but only sometimes. But pure belief in the unknown, by itself, is not crazy. It keeps our eyes open.

And in the end, both science and religion/belief in the unknown serve one identical purpose: they both make us feel very small in the universe.