In the military, is it better to be a person of the letter of the law or the spirit of the law?

It depends. As commander I had a military member essentially go AWOL. As the story unfolded he and his wife had differences on the treatment of their children - so he took them away to protect the kids. The spouse came to see me to order her husband to do what she wanted. The squadron's First Sgt. and I had several discussions on this so I was aware of the situation in general - she had more issues than he did. The military member did call me and explained the situation.
Should I have charged him with being AWOL (the letter of the law), or ignore that to try to help the family? In the end, I decided not to charge the military member since he was looking out after his kids. They later divorced and he took custody of the kids. I wonder what would have happened to him and the family if I put this guy in jail for being AWOL. I think judgement is the best guide, and that comes with experience and maturity. Lacking those, you seek others who have experience and maturity and ask them (noting that its not their decision to make, but ultimately yours). By the way, don't tell anyone I did this - it's a secret.....

I know this will sound like a copout, but I'd say "both". Follow the letter of the law at a bare minimum. However, following the intent of the law will ensure that you never come close to crossing any line. You don't want to be that guy who is trying to find ways to run up to the edge of the cliff, just to see how far you can go. If you do, you just might find yourself at the mercy of a Commander who goes after you using the intent of the law. When in doubt, seek clarification in advance.

Every situation requires leaders to make decisions that are in the best interest of the Army, some being letter of the law and some being intent of the law! It gets easier the longer you are in!