"to a degree" vs. "to an extent"

I'd say: simply style, when not in a geometrical context. Similarity or difference is being metaphorically expressed as distance, whether angular or linear.

Extent means "the amount to which something is or is believed to be the case," and to a degree means "to some extent."

There is no difference between the phrases, except (maybe) the position they take in a sentence.

Everyone will have to compromise to some extent.
They altered the document to such an extent that it contained little in the way of new policy.
To a degree, it is possible to educate oneself.

Here's how I've heard the difference explained: You have a glass of water with a drop of poison. If you drink any of the water, you will die. The degree of poison in the cup is only 1, yet the extent of the poison is 100%.