[Nattybumppo]

The differences are indeed subtle. Here are definitions of the two:

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冷える:
1. To lose heat; to become cold. Also, to feel chilly.
2. (Of an interest or emotion) To go away.

冷める
1. To lose heat; to decrease in temperature.
2. (Of a strong emotion or feeling) To decrease in intensity.

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As you can see, they're pretty similar. However, 冷える seems a bit "colder," and is the verb of choice in sentences like this:

夜は冷えるから、風邪をひかないように！
It gets cold at night, so be careful not to catch a cold!

冷める is used differently; instead of reaching a state of chilly, "brrr" cold like 冷える, 冷める is more often used to describe something reaching a state of thermal equilibrium with its surroundings. It's often used for describing hot liquids cooling to a lukewarm temperature, like in the following sentence:

お茶を冷めないうちに飲みなさい。
Drink your tea before it gets cold.

Note that this usage of "gets cold" in English doesn't actually mean that the drink is getting cold; it means that it's becoming the same temperature as the surrounding air. 冷める is used like this.

Their usages with respect to emotions and interests tend to be more similar than their applications to concrete objects, where the difference between the two verbs is easier to see.

I hope that clears up things a bit.