Information theory provides the ability to measure the amount of entropy in any system. In thermodynamics, it is frequently much easier to derive the entropy values indirectly. Incidentally, DNA is very much related to information theory and is a fine example of mathematical coding in biological systems. Information theory is extraordinarily powerful, but for the practice of engineering one rarely needs to go to that level to get the answer one is looking for.
The greatest danger of NOT tying information theory and thermodynamic entropy together is that people take them to mean different things and ascribe characteristics to thermodynamic entropy that would be clearly invalid if you understood information theory. There is a lot of evidence that this is in fact going on when I see words such as "complexity" to define entropy in thermodynamics, when that is a nonsense definition under the formal information theory definition of entropy. They do not use different definitions; it is just easier to fudge correct results with invalid definitions in thermodynamics than it is in information theory (assuming you aren't doing real work with it, that is).
You are correct about one thing though: "entropy" as it is actually defined is not an easy to grasp or intuitive concept. In this case though, the "made-for-TV" definitions are proving to actually be detrimental to people's understanding of it when they try to apply it.
I am doubtful about this, and perhaps I should explain why. I think the common assumption about DNA, that is somehow runs the show like a computer CPU is terribly distracting. From one point of view, a body is just a conglomeration of carnot cycles that are working desperately to restore themselves to equilibrium, and they do this through a negative feedback op amp with the DNA's ability to produce RNA as the gate of the feedback loop. DNA has inhibitors attached to it that prevent expression by clamping down on the DNA in one end, but can premit expression when the inhibitors catch, or--fail to catch, depending on their particular design--the molecules they were designed for on the other end.
All these cycles operate more or less independently, without a central master control. We are basically a citric acid cycle with some miscellaneous additional cycles as temporary doo-dads attached, most of which are to make up for the inefficiencies of the other doo-dads. What's why this is, was, and always will be a worm planet, with a light sprinkling of evanescent sports on the top.
It would be more accurate, in my opinion, to view DNA as off-line bulk storage. Not as critical, information-rich, structurally controlling content. DNA is inert crystals if you ask it to run the show. It does not run the show by communicating information as Shannon understood information, so it is, in my opinion, irrelevant in discussions of evolutionary fitness, to ask after the metrics of the information entropy it exhibits.