If I've given the impression that the difference is merely a question of varying encodings, then I have to agree my arguments have thus far been lacking.

The idea that a genome as expressed in nucleic acid is purely, and only, an informational medium, is fundamentally in error. It does encode information in the sequence of base pairs, this is true. But it is also a physical structure in its own right, and properties of that structure incidental to the encoded information have what recently looks to be at least as important a role in the process of transcription as the sequence itself.

There are, for example, some sequences which will cause a ribosome to transcribe the surrounding genes differently or with varying frequency, due to the physical interaction between the molecules involved. (I recently discussed this here in the context of recent research on causes of eye color; it should not be too far back in my comment history.) We also see, for example, that both viral and eukaryotic DNA can be and often are transcribed in ways that produce different proteins from the same sequence, again as a result of physical constraints affecting the interaction with the ribosome. This is one reason why "junk DNA" is a bit of a misnomer, and why we more recently see the term fall out of use in favor of "noncoding DNA" - these regions carry no information in their own right, but nonetheless can strongly affect the outcome of transcription because transcription is not only an informatic process. This isn't true of software; there is no general case in which two programs varying only in nonsyntactic ways will be evaluated differently under otherwise identical conditions - we create programming languages as we do in part to ensure that won't happen, and it's also part of the reason why we use transistors instead of vacuum tubes or relays: in order to engineer that kind of variance as much as we can out of existence. What is therefore an accidental property in software is an essential one in gene expression, and cannot be overlooked without reaching an inaccurate conception of how the latter process works.

That's just one example, and it's true that processes like these can be modeled in software to variously imperfect degrees of fidelity and that information-theoretical models can be useful in understanding some aspects of how they work. But that's not the same thing as them working similarly enough that understanding one very well suffices to reason about the other. I definitely can see how it's easy to assume otherwise! It's an assumption I shared, before my own yearlong exposure to the field at a sufficient level of detail to start to understand what I hadn't understood about it before, and considerable reading and study thereafter.

Unfortunately, I was there to provide engineering support to people doing that work, not to do it myself, and the knowledge I've derived from that experience apparently does not extend so far as producing a concise and positive statement of the fundamental difference between the two fields of study - I spent considerably more time teaching informaticists how to program, formally and otherwise, than I spent learning about bioinformatics. That leaves me able to recommend little beyond seeking out similar experience of your own, which I do recommend if the depth of your interest suffices -although I do also have to say working in academia as a nonacademic has very little else to recommend it.

I know there are some folks on HN with formal knowledge and training greatly exceeding my own, and some of whom have probably also had experience teaching the basics in an accessible way. Perhaps one of them might give a more useful answer here than I've been able to.