Perhaps the common denominator is "problem solving."
Beyond that, i doubt i doubt i could provide any insight, but i can at least provide a limited answer from personal experience.
This issue arises for us in hiring--i.e., do we hire a programmer and teach them statistics or do we hire a statistics person and teach them to program? Ideally we could find someone fluent in both discipline, and indeed, that's the third net we cast, but rarely with any success.
Here's an example. The most stable distinction between the two activities (software dev & statistical analysis) is probably their respective outputs, or project deliverables. For instance, in my group someone is conducting the statistical analysis on the results of our split-path and factorial experiments (e.g., from the t-test results, whether the difference is significant, or whether the test ought to continue). That analysis will be sent to the marketing department which they'll use to modify the web pages comprising the Site with a view towards improving conversion. A second task involves the abstraction of and partial automation of those analyses so the results can be processed in near-real time.
For the first task, we'll assign a statistician; for the second, a programmer. The business problem we are trying to solve is the same for both tasks, yet for the first, the crux is statistics, for the second, the statistics problems have been largely solved and the crux is a core programming task (I/O).
Notice also how the evolution of the tools associated with the two activities have evolved so the distinction between the two (software dev & data analysis) is further obfuscated: mainstream development languages are being adapted for use as domain-specific analytical tools, at the same time, frameworks continue to be developed which enable the non-developers to quickly build lightweight, task-oriented applications in DSLs.
For instance, python, a general purpose development language has R bindings (RPy2) which along with its native interactive interpreter (IDLE), substantially facilitates Python's use in statistical analysis, while at the same time, there is a clear trend in R package development toward (web) application development: R Bindings for Qt, gWidgetsWWW, and RApache--are all R Packages directed to Client or Web App development, and whose initial release was (i think) w/in the past 18 months. Aside from that, since at least the last quarter of last year, i've noticed an accelerating frequency of blog posts, presentations, etc. on the subject of Web app development in R.
Finally, i wonder if your question is perhaps evidence of the growing popularity of R. Here's what i mean. A decade ago, when my employer purchased a site license, i began learning and using one of the major statistical computing products (no point here in saying which one, it begins with "S"). i found it unnatural and inflexible. Unlike Perl (which i was using at the time) this tool was not an extension of my brain (which isn't an optional attribute of an analytical tool, to me it's more or less the definition of one). Interacting with this System was more like using a vending machine--i selected some statistical function i wanted and then waited for the "output", which was often an impressive set of high-impact, full-color charts and tables. Nearly always though what i wanted was to modify my input or use that output for the next analytical step. That seemed to required another, separate trip to the vending machine. The fact that this tool was context-aware--i.e., it knew statistics--while Perl didn't, didn't compensate for the awkward interaction. Statistical analysis done this way would never be confused with software development. (Again, this is just a summary of my own experience, i don't claim it can be abstracted. It's also not a polemic against any (or all) commercial data analysis platforms--millions use them and they've earned zillions for the people who created them, so let's assume it was my own limitations that caused the failure to bond.)
I had never heard of R until about 18 months ago, and i only discovered it while scanning PyPI (The Web Interface to Python's external package repository) for statistics libraries for python. There i came across RPy, which seemed brilliant but required a dependency called "R" (RPy of course is really just a set of Python bindings to R).
Perhaps R appeals to programmer and non-programmers equally, still for a programmer/analyst, this was a godsend. It hit everything on my wish list for a data analysis platform: an engine based on a full-featured, general programming language (which in this case is a proven scheme descendant), an underlying functional paradigm, built-in interactive interpreter, native data types built from the ground up for data analysis, and the domain knowledge baked in. Data analysis became more like coding. Life was good.
