Why does the verb "overlook" have such a different meaning from "oversee"?

Some basis here:

Just because you look (in the sense of pointing your eyes in the objects general direction) at something does not mean you see (in the sense of "to perceive") it.

When you look at a forest, do you see each single tree? Probably not.

Also, see has the idea built in of looking at a detail, whereas look has more of a passive action attached to it.

Back to the question:

In the case of overlook, you are looking above (ie: over) something, so you aren't looking at it. Looking above (ie: over) an object makes you miss seeing it - your eyes are not pointed at the object.

In the case of oversee, the concept is that you are standing (conceptually if not physically) over whatever you are see-ing, and making sure the object (usually a person) actually does what it is supposed to be doing.

In the past they have been used with each other's meanings, but these senses have now fallen out of use. Just part of the way language changes - they used to overlap more, but now they don't, and there may not be any particular reason why.

An example from a letter of George Washington to Alexander Spotswood (24 September 1798):

I shall depend upon your Overseer, Roger Farril, for my Mansion house concern; and Brookes Joiner, to overlook my Carpenters.

Scanning through the rest of the letters, he seems to use "overlook" for "oversee" fairly often - even, as here, in the same letter.

And for the reverse, from the play The Way of the World by William Congreve (1700):

'Twas for my ease to oversee and wilfully neglect the gross advances made him by my wife, that by permitting her to be engaged, I might continue unsuspected in my pleasures, and take you oftener to my arms in full security.

Neither of these senses are current in English any more.