I know that UV index is not the same thing as brightness but it often seems that the UV index is low when it's overcast and is high when it's clear and sunny.

Hi, this may help:
What is the UV index and how is it calculated? | HowStuffWorks

UV index, Ultra Violet Index, is basically a ratio of how much UV hits the earth, depending on where you live, and how much could hit the earth. It is typically calculated at Noon since you could have a very bright sky at 6PM with little UV Index. That is why they tell most people to stay indoors from 11-3.. Clear sky may or may not have a bearing depending on your Latitude. Living on the Equator makes a big difference over lets say, Alaska since the angle of the sun is very different.