These are such depressing words, but DD, in it's infinite wisdom, thought indigence was a good wotd, and that led to a natural comparison with destitute. At least indigence wasn't the wotd for April 15th...
Indigence derives from the Latin "indu" for "in or within" and "egere" for "to be in need or want", which appears to be a little redundant, but fortunately, since the meaning is still "in need", I don't need to go into a lengthy analysis of the etymology or the derivation. Perhaps the double "in" just gives greater emphasis to the state of need which makes it a "seriously impoverished condition" and not merely a basic "in need".
Destitute, on the other hand, derives from the Latin "de" as a form of "dis" for "out, off, apart, away, completely" and "stit" for "place or put". Therefore, destitute means "out of place" and by reasonable extension, without the comforts associated with place (shelter, clothing, food). Of course, then next natural and perhaps lazy association was just simply to be without something whether a basic necessity or not.
So, of course, this leaves us with two words which still appear to have the same meaning, of being without something necessary. Time for OED. Well, OED does shed some light. Destitute has a connotation of having been abandoned or deprived, and therefore, that the circumstances were brought on by another person, while indigence is from the personal perspective of being in want or need, not necessarily due to the actions of another person, and almost exclusively relates a lack of money as due, since with money we could fulfill these basic wants and needs. When she was evicted from her apartment, she became destitute, but she could not pay the rent because of her indigence from choosing not to work. Now, because we have to push the boundaries... Destitute can be used with any living being, but indigence from a standpoint of wanting and needing a basic necessity only applies to people. Standing on the T platform in the light snow, clutch her coat against the wind and half asleep after having missed her stop some 4 stops earlier, she looked destitute, and in pity, the T driver stopped to pick her up even though this was not the inbound platform. True story. When he was downsized from his place of employment of the last 30 years, he looked destitute and confused, but fortunately his 401k ensured that he would not be indigent. The lone obsolete 386 computer looked destitute on a table of Intel Core Duos. Possibly, it you think computers have basic needs (talk to Pixar), but this also goes to the original meaning of being abandoned and out of place, notwithstanding that it was intended for people. Many graduate students are indigent which is an unintended tax classification for living below the poverty line. Beggars on the streets of Cape Town are indigent but not destitute since they have their Township home. Animals in a humane society pull at our heart strings because they are destitute. I can't write any more of these "destitute" sentences--they make me sad. Ok, well, maybe one more. She smiled as the disbarred Plaintiff's counsel left his office, now destitute, and likely to become indigent as a result of being unemployable.
May you never be destitute nor indigent.