Is a Harvard PhD in applied mathematics (from engr. dept.) as reputable as a PhD in pure mathematics (from math dept)? Most rankings systems rank math at Harvard quite favorably. Is phd in applied mathematics (control theory) at Harvard as reputable as a phd in pure math at Harvard? (The first is offered in an engineering department while the second is offered in a math department)
I got my PhD at Harvard in mathematics.  Not in "pure math" but just in "math."  Harvard does not have an "applied mathematics" department or a "PhD program in applied mathematics."  Rather, applied mathematics is one of seven "areas of study" in the school of engineering and applied sciences: see here.  What does this mean?  Well, see here.  It says:


  In keeping with the interdisciplinary nature of modern research, we do not have traditional academic departments and do not award degrees by specific research area. Instead, graduate students work toward a degree in one of seven subjects—Applied Mathematics, Applied Physics, Computer Science, Computational Science &amp; Engineering, Data Science, Design Engineering, and Engineering Sciences.


Some of these subjects have "PhD model programs," but when you click on applied math, you find


  Due to its highly interdisciplinary nature, Applied Mathematics does not have a specific model program.  Students pursuing a Ph.D. in Applied Mathematics should consult with their Director of Graduate Studies: Professor Ariel Amir (fall), or Professor Yue Lue (spring), concerning their degree programs.


Interestingly, Professor Amir is an Assistant Professor of Applied Mathematics.  [Interesting in part because the mathematics department has had only "full" professors for at least 20 years.]  Here is a listing of faculty in applied mathematics at Harvard: the only other professor who is listed as being solely in applied mathematics is also an assistant professor.  This suggests to me that applied math at Harvard may be in the process of becoming more of an academic department than it has been or is now.  The time would be ripe, in view of the record $400 million gift of Mr. Paulson in 2015.

All that was mostly background.  Now I can tell you: when I was at Harvard (1998-2003) there was almost no direct contact between the mathematics department and the mathematics part of SEAS: I knew one SEAS PhD student who was (for some reason) teaching freshman calculus along with the math students.  (I just looked him up: he has what looks like a lucrative industry job in Boston.)  So the answer to your question is: No one I know is comparing these two programs. They are different in every way, with almost no overlap.  It's not as though you could choose to do control theory in the math department at Harvard: there is no one there that does that.  (There is one professor in the Harvard math department who does what I at least would call applied mathematics: Martin Nowak, the Professor of Biology and Mathematics and Director of the Program for Evolutionary Dynamics.)

It would make more sense to compare the engineering program at Harvard to other engineering programs.  US News ranks Harvard's engineering program at #23 currently.  If you look at this page it gives you the ranking of a variety of programs at Harvard, and actually it has the lowest rank of the more than 100 programs and subprograms listed there.  The math department is currently ranked third [it has been ranked in the top three for many years].  So maybe that is a sense in which "pure math is more reputable," although again I stress that this does not seem like the right way to look at it.