Differences between German computer science doctorates
In general, the latin title of the degree conferred at a German university is of purely historical relevance (basically, indicating how the original four faculties of the middle ages split into the dozen or more departments of today, which happened in different ways at different universities -- obviously, Computer Science split off of Electrical Engineering in Darmstadt, rather than, say, Mathematics as in Munich); the important information is which department conferred the degree (which differentiates a Dr.rer.nat in Mathematics from one in Physics or Chemistry).

There is, however, one exception: The Dr.-Ing. (Doctorate in Engineering) still has a special place as an indication of the high regard in which German academic engineering is held (mostly, needless to say, within Germany, as well as by those directly influenced by this system, although a hundred years ago this was indeed something special), and is the reason why Darmstadt is offering a choice -- in effect, it's a marketing strategy to get students to enrol by offering them a degree that's perceived as more attractive. (This is also related to the fact that some universities still hand out "Dipl.-Ing." diplomas even though the Bologna process turned every other degree into a Masters.)

As to whether the choice of degree will make a practical difference in finding a job, that's hard to say. My feeling is that it will most likely play a role when transitioning to industry in Germany, where someone with a Dr.-Ing. might be seen as more "hands-on", practical-oriented, than with a Dr.rer.nat. (especially at small or medium-sized companies which might hesitate to hire a purely theoretical PhD). On the German academic job market, on the other hand, the university (and research group) where you got your title would be much more important. Outside Germany, I'd be surprised if many people cared much about the distinction (unless they themselves are the proud holders of a Dr.-Ing., of course...)