It's not the easy answers I'm looking for like, both designed computers; Babbage the Analytical Engine and Turing the Pilot ACE. Or that both were mathematicians, or even both were obviously Englishmen.
The answer is... both were wartime codebreakers. Turing is of course celebrated for cracking German Naval Enigma whilst at Bletchley Park during WWII. He's also less well known for contributing to cracking the much more complex German High Command Lorenz code.
What you probably didn't know is that Charles Babbage cracked the Vigenère cipher, which was a form of polyalphabetic substitution cipher (like Enigma) that had been in widespread use since the 16th century. Babbage is reputed to have cracked the cipher as early as 1846, but he didn't publish his method because the British were soon to be at war with Russian in the Crimea and the Russians were known to use the Vigenère cipher. Thus, as in WWII, the British would be able to read their enemies' intercepted messages and gain a strategic advantage. All of this history is my book The Universal Machine.