The question of "why were physicists targeted" is made clear enough in one of the other answers.* The "why did rocket engineers prosper" is a somewhat different answer, though, and I will try to address it briefly.

*That other answer, for whatever reason, was deleted. The long and short of it is that many of the theoretical physicists working in Germany were Jewish — they were over-represented. The reason for this is that experimental physics was considered the real "prize" of physics in Germany in the late-19th and early-20th centuries. Jews were discriminated against to a degree (not as much as under the Nazis, of course), and found it very hard to establish themselves as experimentalists. As a result, many ended up going into theoretical pursuits. Like many demographic cycles this became somewhat culturally reinforced over time. This is not to say that there were not Jewish experimentalists — there were. But in the areas of theory, they were extremely visible. Anyway, there was briefly an anti-"Jewish physics" movement (Deutsche Physik, which translates as "German physics" but could be also considered "True-German physics" or "Aryan physics" — the "German" here is meant to be excluding Jews and to a lesser extent even the British and French contributions). As my other comment makes clear I think the effects of this have been a bit exaggerated (it was never a platform of the National Socialist party, and it came "from below," in the sense of it coming from several members of the German physics community, not "from above," in the sense of from the Party, and it was not especially successful in any event). Anyway, the long-and-short of it is that the theoretical physicists were targeted because a healthy number of them were Jewish, and Jews in general were targeted. Jews in academia were targeted especially early on, because the German educational system is part of the civil service, and the first major piece of anti-Semitic legislation was about "reforming" the civil service (which involved firing all of the Jews in it). So Jewish academics in general began leaving Germany very early on, circa 1933 and even in 1932 in some instances.

First, it is important to note that the Nazis were more inclined towards engineering than they were towards basic science in any event. This is a broad statement, to be sure. But in the choices and themes the Nazis took up, one finds that they had very little problem with technology — which they thought could be mobilized towards whatever ends they wanted — and some major issues with science — which could interact with what they thought were political questions in an unfavorable way. The Nazis, for example, thought that biology was core to their political platform, as it served as the underlying ideology for how the state worked. There was no such connection with aerospace engineering. Nothing that aerospace engineers could do or say would cast into doubt the Nazi political platform. Biology and medicine could, by contrast, so they had to be much more carefully dealt with. (Whether physics did or didn't, I discuss in my comment on Deutsche Physik elsewhere in this thread.) There are more connections than this, but it is just worth considering that there was a difference, however fuzzy of one, between "science" and "engineering" in this context. (On the Nazi attitudes towards science and technology, see Jeffrey Herf, Reactionary Modernism.)

Second, rocket work in particular has a very interesting history in interwar Germany. Rockets were not prohibited by the Treaty of Versailles, unlike many other forms of armament. So work on rockets could be done without any fear of discovery. Early work on rockets (in many countries) was deeply tied to fantasies about space exploration. This resonated very strongly with the Nazis, who liked this sort of long-term, big-ambition thinking. (It also resonated with the Russians and Americans, but that is a separate story.) And, lastly, it is well-known that Hitler was himself quite fond of the idea of new and fantastical "wonder weapons" that he thought would cause his enemies to immediately surrender. (He was not alone in this approach — Winston Churchill had the same affectations. It was a common-enough response of the generation of military and political leaders who saw the devastation that technology had wrought in the First World War. )

The result of all of this is that almost immediately after the Nazis took power they made rocket work an official area of sponsorship by the Army. It hit many of their "buttons" — it was good, "German" practical technology (as opposed to "idle theorizing" that they associated with Jewishness), it also served the aims of rearming after Versailles, it also served as effective propaganda ("wonder weapons"), it also played into their long-term ambitions about the human race (space travel). It was, in reality, a terrible decision for them to have made: the V-1 and V-2 programs were ruinously expensive, did very little damage compared to the resources that were expended on them (more people died building the rockets than died as a consequence of their use), and did not, in the end, cause the Allies to surrender.

One last comment: nuclear fission was not discovered until 1939. Until then, nuclear physics had no military applications whatsoever. Even after that point, it was not clear that nuclear weapons could be built at all, or in the very short term. The German scientists correctly concluded that it would be very, very hard to build nuclear weapons — there was a lot of risk involved, it was unlikely they would be decisive during the war. And, indeed, the US and UK barely managed to build an atomic bomb by the end of the war (they did not build any in time to be used in the European war, and if they had been a few months later in making them — no difficult thing to imagine — they wouldn't have had them ready for the Pacific war either). So while the Germans made many technical errors in the war (e.g. over-investing in rocketry), I do not really consider their lack of investment in nuclear weapons to be one of them. Knowing what we know now about the difficulty in making them, and the additional difficulties the Germans would have faced (because they were subject to aerial bombings, unlike the Americans), it seems very unlikely they actually would have been able to build nuclear weapons during the war even if they had someone concluded that was something they ought to be expending more resources on. This is separate from the fact that they had a huge amount of "brain drain" as a result of their anti-Semitic policies, which wouldn't have helped. (On the German nuclear program, Mark Walker's Nazi Science is very readable and goes into many of the administrative and technical angles in an intelligent way.)

Obviously Einstein and Bohr were Jews so they left early

"Obviously"? Bohr was Jewish, yes, but he did not leave Nazi Germany. This is not possible, since he didn't live in Nazi Germany!

Niels Bohr founded the Institute of Theoretical Physics at University of Copenhagen in 1920. He lived in Denmark until 1943, at which point he fled to England (by way of Sweden) based on a tip that he was about the be arrested by the German occupation force.

This is all part of public record, so I hope I am not speaking out of term wrt. the /r/AskHistorians rules. But when you state such a fallacy prefaced with "Obviously", I can hardly keep silent.

This isn't an answer to the question, but I feel it's important to add that german mathematics suffered in a similar way as theoretical physicists. For example, to paraphrase, David Hilbert was asked by minister of education Bernhard Rust if the removal of Jewish mathematicians from the University of Göttingen truly had such a negative impact on the mathematical institute. He essentially replied that it had not only suffered, but that it didn't exist anymore. He didn't mean it literally of course, but said it to express how much mathematics at University of Göttingen had suffered. The mathematicians forced out by the nazi party included Hermann Weyl, Emmy Noether, and Felix Bernstein. There were certainly some mathematicians such as Oswald Teichmüller who supported the Nazis, but I think the comment by Hilbert gives a good indicator of what happened at the University of Göttingen at the very least.